THE CLASSIFICATION OF p-COMPACT GROUPS FOR p ODD

KASPER K. S. ANDERSEN, JESPER GRODAL, JESPER M. MØLLER, AND ANTONIO VIRUEL

Contents

1. Introduction 1 Relationship to the Lie case and the conjectural picture for p = 2 7 Organization of the paper and notation 7 Acknowledgments 8 2. The map Aut(BX) → Aut(BN_{X}) 8 3. Automorphisms of maximal torus normalizers 9 4. Reduction to connected, center-free simple p-compact groups 12 5. An integral version of a theorem of Nakajima and realization of p-compact

groups 18 6. Proof of the main theorem using Sections 8, 9, 10, 11, and 12 21 7. Consequences of the main theorem 32 8. Elementary abelian subgroups of the exceptional groups 38 8.1. Recollection of some results on linear algebraic groups 39 8.2. The groups E_{6}(C) and 3E_{6}(C), p = 3 42 8.3. The group E_{8}(C), p = 3 52 8.4. The group 2E_{7}(C), p = 3 59 9. Non-toral elementary abelian p-subgroups of projective unitary groups 62 10. Calculation of the obstruction groups 63 10.1. The toral part 64 10.2. The non-toral part for the exceptional groups 66 10.3. The non-toral part for the projective unitary groups 69 11. Appendix: The classification of finite Z_{p}-reflection groups 71 12. Appendix: Invariant rings of finite Z_{p}-reflection groups, p odd (following

Notbohm) 76 13. Appendix: Outer automorphisms of exotic finite Z_{p}-reflection groups 79 References 82

1. Introduction

It has been a central goal in homotopy theory for about half a century to single out the homotopy theoretic properties characterizing compact Lie groups, and obtain a corresponding classification, starting with the work of Hopf [68] and Serre [113, §4] on H-spaces and loop spaces. Materializing old dreams of Sullivan [123, p. 5.96] and Rector [111], Dwyer and Wilkerson, in their seminal paper [48], introduced the notion of a p-compact group, as a suitably “finite” p-complete loop space, and proved that p-compact groups have many Lie-like properties. Even before their introduction it has been the hope [110], and later the conjecture [51, 80, 41], that these objects should admit a classification much like the classification of compact connected Lie groups, and the work toward this has been carried out by many authors. The goal of this paper is to complete the proof of the classification theorem for p an odd prime, showing that there is a one-to-one correspondence between connected p-compact groups and finite reflection groups over the p-adic integers Z_{p}. We do this by providing the last—and rather intricate—piece, namely that the p-completion of the exceptional compact Lie groups are uniquely determined as p-compact groups by their Weyl groups, seen as Z_{p}-reflection groups. In fact we will give an essentially self-contained proof of the entire classification theorem.

The first named author was supported by EU grant EEC HPRN-CT-1999-00119.

The second named author was supported by NSF grant DMS-0104318, a Clay liftoff fellowship, and the Institute for Advanced Study for different parts of the time this research was carried out.

We start by very briefly introducing p-compact groups and some objects associated to them necessary to state the classification theorem—we will later in the introduction return to the history behind the various steps of the proof. We refer the reader to [48] for more details on p-compact groups and also recommend the overview articles [41, 80, 88]; we likewise point out that it is the technical advances on homotopy fixed points by Miller [84], Lannes [79], and others which makes this theory possible.

A space X with a loop space structure, for short a loop space, is a triple (X, BX, e) where BX is a pointed connected space, called the classifying space of X, and e : X → ΩBX is a homotopy equivalence. A p-compact group is a loop space with the two additional properties that H^{∗}(X; F_{p}) is finite dimensional over F_{p} (to be thought of as “compactness”) and that BX is F_{p}-local [16][48, §11] (or, in this connection, equivalently F_{p}-complete [17]). Often we refer to a loop space simply as X. When working with a loop space we shall only be concerned with its classifying space BX, since this determines the rest of the structure— indeed, we could instead have defined a p-compact group to be a space BX with the above properties. The loop space (G_{p}^{∧}, BG_{p}^{∧}, e), corresponding to a pair (G, p) (where p is a prime, G a compact Lie group with component group a finite p-group, and (·)_{p}^{∧} denotes F_{p}-completion [17][48, §11]) is a p-compact group. (Note however that the Lie group G is not uniquely determined by BG_{p}^{∧}, since we are only focusing on the structure “visible at the prime p”; e.g., B SO(2n + 1)_{p}^{∧} ≃ B Sp(n)_{p}^{∧} if p ≠ 2, as originally proven by Friedlander [59]; see Theorem 11.4 for a complete analysis.)

A morphism X → Y between loop spaces is a pointed map of spaces BX → BY. We say that two morphisms are conjugate if the corresponding maps of classifying spaces are freely homotopic. A morphism is called an equivalence if it has an inverse up to conjugation, or in other words if BX → BY is a homotopy equivalence. If X and Y are p-compact groups, we call a morphism a monomorphism if the homotopy fiber of the map BX → BY is F_{p}-finite.

The loop space corresponding to the p-completed classifying space BT = (BU(1)^{r})_{p}^{∧} is called a p-compact torus of rank r. A maximal torus in X is a monomorphism T → X such that the homotopy fiber of BT → BX has non-zero Euler characteristic. (We define the Euler characteristic as the alternating sum of the F_{p}-dimensions of the F_{p}-homology groups.) Fundamental to the theory of p-compact groups is the theorem of Dwyer-Wilkerson [48] that, analogously to the classical situation, any p-compact group admits a maximal torus. It is unique in the sense that for any other maximal torus i′ : T′ → X, there exists an equivalence φ : T → T′ such that i′φ and i are conjugate. (Note that there is a subtle difference between this statement and the classical statement of being “unique up to conjugation” due to the fact that a maximal torus is defined to be a map and not a subgroup.)

Fix a p-compact group X with maximal torus i : T → X of rank r. Replace the map Bi : BT → BX by an equivalent fibration, and define the Weyl space W_{X}(T) as the topological monoid of self-maps BT → BT over BX. The Weyl group is defined as W_{X}(T) = π_{0}(W_{X}(T)) [48, Def. 9.2]. By [48, Prop. 9.5] W_{X}(T) is a finite group of order χ(X/T). Furthermore, by [48, Thm. 9.7], if X is connected then W_{X}(T) identifies with the set of conjugacy classes of self-equivalences φ of T such that i and iφ are conjugate. In other words, the canonical homomorphism W_{X}(T) → Aut(π_{1}(T)) is injective, so we can view W_{X}(T) as a subgroup of GL_{r}(Z_{p}), and this subgroup is independent of T up to conjugation in GL_{r}(Z_{p}).

Now, by [48, Thm. 9.7] this exhibits (W_{X}, π_{1}(T)) as a finite reflection group over Z_{p}. Finite reflection groups over Z_{p} have been classified for p odd by Notbohm [95] extending the classification over Q_{p} by Clark-Ewing [29] (which again builds on the classification over C of Shephard-Todd [115]); we recall this classification in Section 11 and extend Notbohm’s result to all primes. (Recall that a finite Z_{p}-reflection group is a pair (W, L) where L is a finitely generated free Z_{p}-module, and W is a finite subgroup of Aut(L) generated by elements α such that 1−α has rank one viewed as a matrix over Q_{p}; two finite Z_{p}-reflection groups (W, L) and (W′, L′) are called equivalent, if we can find a Z_{p}-linear isomorphism φ : L → L′ such that the group φW φ^{−1} equals W′.)

The main classification theorem which we complete in this paper, is the following.

Theorem 1.1. Let (W, L) be the assignment which to each connected p-compact group X assigns the Weyl group W_{X}, seen as a Z_{p}-reflection group via the action of W_{X} on its Z_{p}-lattice L_{X} = π_{1}(T). If p is an odd prime, then the assignment (W, L) gives a bijection between the set of connected p-compact groups up to equivalence and the set of finite Z_{p}- reflection groups up to equivalence.

Furthermore the group of outer automorphisms of X (that is the group of unpointed homotopy classes of self-homotopy equivalences of BX) is naturally isomorphic to the outer automorphism group of (W_{X}, L_{X}), that is N_{GL(L_{X})}(W_{X})/W_{X}. (See Lemma 2.1 for the map giving this equivalence of automorphism groups, and Section 13 for a calculation of these.)

In particular this proves, for p odd, Conjecture 5.3 in [41] (see Theorem 1.4). The selfmap part of the statement can be viewed as an extension to p-compact groups, p odd, of the main result in [75], which again extends [74]. (Our method of proof is via centralizers and so “dual”, but logically independent, of the one in [74, 75] (see e.g. [40, 65]).)

By inspection of the classification of finite Z_{p}-reflection groups one sees that the theorem has as a corollary that the theory of p-compact groups on the level of objects splits in two parts, as has been conjectured (Conjecture 5.1 and 5.2 in [41]).

Theorem 1.2. Let X be a connected p-compact group, p odd. Then X can be written as a product of p-compact groups

X ≅ G_{p}^{∧} × X′

where G is a compact connected Lie group, and X′ is a direct product of exotic p-compact groups.

(A p-compact group is called exotic if (W, L ⊗ Q) is irreducible and does not come from a Z-reflection group; see Section 11 for more information.) Theorem 1.1 has both an existence and a uniqueness part to it, the existence part being to realize all possible finite Z_{p}-reflection groups as Weyl groups of connected p-compact groups. The finite Z_{p}-reflection groups which come from compact Lie groups are of course realizable, and the finite Z_{p}-reflection groups where p does not divide the order of the group can also relatively easily be dealt with, as done by Sullivan [123, p. 5.96] and Clark-Ewing [29] long before p-compact groups were officially defined. The remaining cases were realized by Quillen [108, §10], Zabrodsky [134], Aguadé [3] and Notbohm-Oliver [102] [97, Thm. 1.4]. (The classification of finite Z_{p}- reflection groups guarantees that the construction of these examples actually enables you to construct all finite Z_{p}-reflection groups as Weyl groups of connected p-compact groups.)

The work toward the uniqueness part, to show that a p-compact group is uniquely determined by its Weyl group, also predates the introduction of p-compact groups. The quest was initiated by Dwyer-Miller-Wilkerson [56, 44] (building on [1]) who proved the statement (using slightly different language) in the case where p is prime to the order of W_{X} as well as for SU(2)_{2}^{∧}. Notbohm [100] and Møller-Notbohm [87, Thm. 1.9] extended this to a uniqueness statement for all p-compact groups X where Z_{p}[L_{X}]^{W_{X}} (the ring of invariant polynomial functions on L_{X}) is a polynomial algebra and (W, L) comes from a finite Z-reflection group. Notbohm [102, 97] subsequently also handled the cases where (W, L) does not come from a finite Z-reflection group. (Beware that in [102, 97] Notbohm used a apparently weaker uniqueness statement, from which the above statement however can be deduced; see Remark 7.11.)

To get general statements beyond the case where Z_{p}[L_{X}]^{W_{X}} is a polynomial algebra, i.e., to attack the cases where there exists p-torsion in the cohomology ring, the first step is to reduce the classification to the case of simple, center-free p-compact groups. (A p-compact group X is called simple if L_{X} ⊗ Q is an irreducible W -representation and that it is centerfree is for p odd equivalent to that (L_{X} ⊗Z/p^{∞})^{W} = 0.) The results necessary to obtain this reduction was achieved by the splitting theorem of Dwyer-Wilkerson [50] and Notbohm [99] along with properties of the center of a p-compact group established by Dwyer-Wilkerson [49] and Møller-Notbohm [86]. (We carry out this reduction in Section 4; see also [89].)

An analysis of the classification of finite Z_{p}-reflection groups together with explicit calculations (see [96] and Theorem 12.2) shows that, for p odd, Z_{p}[L_{X}]^{W_{X}} is a polynomial algebra for all irreducible center-free finite Z_{p}-reflection groups except the reflection groups coming p-compact groups P U(n)_{p}^{∧}, E_{8}_{5}^{∧}, F_{4}_{3}^{∧}, E_{6}_{3}^{∧}, E_{7}_{3}^{∧}, and E_{8}_{3}^{∧}. (For exceptional compact Lie groups the notation E_{6} etc. denotes their adjoint form.)

The case P U(n)_{p}^{∧} was handled by Broto-Viruel [19] (using a Bockstein spectral sequence argument to deduce it from the result for SU(n)), generalizing earlier partial results of Broto-Viruel [20] and Møller [85]. The remaining step in the classification is hence to handle the exceptional compact Lie groups, in particular the problematic E-family at the prime 3, and this is what is carried out in this paper. (The fourth named author has also given alternative proofs for F_{4}_{3}^{∧} and E_{8}_{5}^{∧} in [126] and [125].)

Theorem 1.3. Let X be a connected p-compact group, for p odd, with Weyl group equal to (W_{G}, L_{G} ⊗ Z_{p}) for one of the pairs (F_{4}, p = 3), (E_{8}, p = 5), (E_{6}, p = 3), (E_{7}, p = 3), or (E_{8}, p = 3). Then X is equivalent, as a p-compact group, to the p-completion of the corresponding exceptional group G.

We will in fact give an essentially self-contained proof of the entire classification Theorem 1.1, since most of the work is necessary anyway for setting up the proof for the exceptional cases. We however still rely on the classification of finite Z_{p}-reflection groups (see [95, 96] and Section 11 and 12) as well as the above mentioned structural results from [48, 49, 86, 50, 99, 90].

The main ingredient in handling the exceptional groups is to get sufficiently detailed information about their many conjugacy classes of elementary abelian p-subgroups (carried out in Section 8, expanding on the work of Griess [63]), and then to use this information to show the relevant obstruction groups are trivial (carried out in Section 10), using formulas of Oliver [103] (see also [65]).

It is possible to formulate a more topological version of the uniqueness part of Theorem 1.1 which holds for all p-compact groups, not necessarily connected, which is however easily seen to be equivalent to the first one using [5, Thm. 1.2]. It should be viewed as a topological analog of Chevalley’s isomorphism theorem for linear algebraic groups (see [70, §32] [122, Thm. 1.5] and [35, 106, 101]). To state it, we define the maximal torus normalizer N_{X}(T) to be the loop space such that BN_{X}(T) is the Borel construction of the canonical action of W_{X}(T) on BT. Note that by construction we have a morphism N_{X}(T) → X, and that this map is independent of the choice of T, up to conjugacy. By [48, Prop. 9.5], W_{X}(T) is a discrete space so BN_{X}(X) has only two non-trivial homotopy groups and fit into a fibration sequence BT → BN_{X}(T) → BW_{X}. (Beware that in general N_{X}(T) will not be a p-compact group since its group of components W_{X} need not be a p-group.)

Theorem 1.4 (Topological isomorphism theorem for p-compact groups, p odd). Let p be an odd prime and let X and X′ be p-compact groups with maximal torus normalizers N_{X} and N_{X′}. Then X ≅ X′ if and only if BN_{X} ≃ BN_{X′}.

Furthermore the spaces of self-homotopy equivalences Aut(BX) and Aut(BN_{X}) are equivalent as topological monoids. Explicitly, turn f : BN_{X} → BX into a fibration which we will again denote by f, and let Aut(f) denote the topological monoid of self-homotopy equivalences of f. Then the following canonical zig-zag, given by restrictions, is a zig-zag of equivalences:

Aut(BX) ←^{≅} Aut(f) →^{≅} Aut(BN_{X}).

(In the above theorem, the fact that the evaluation map Aut(f) → Aut(BX) is an equivalence follows by a short general argument (Lemma 2.1), whereas the equivalence Aut(f) → Aut(BN_{X}) requires a detailed case-by-case analysis.)

We point out that the classification of course gives easy, although somewhat unsatisfactory, proofs that many theorems from Lie theory extends to p-compact groups, simply using the models provided via Theorem 1.2 and using that the theorem is known to be true in the Lie type case, and a case checking in the remaining cases. Since all the exotic p-compact groups have cohomology ring a polynomial algebra, this can turn out to be rather straightforward. In this way one for instance sees that Bott’s celebrated result about the structure of G/T [12] still holds true for p-compact groups, at least on cohomology.

Theorem 1.5 (Bott’s theorem for p-compact groups). Let X be a connected p-compact group, p odd, with maximal torus T. Then H^{∗}(X/T; Z_{p}) is a free Z_{p}-module of dimension |W_{X}|, concentrated in even degrees.

Likewise combining the classification with a case-by-case verification for the exotic groups by Castellana [24, 25], we get that the Peter-Weyl theorem holds for connected p-compact groups, p odd:

Theorem 1.6 (Peter-Weyl theorem for connected p-compact groups). Let X be a connected p-compact group, p odd. Then there exists a monomorphism X → U(n)_{p}^{∧} for some n.

We also still have the ‘standard’ formula for the fundamental group (the subscript denotes coinvariants).

Theorem 1.7. Let X be a connected p-compact group, p odd. Then

π_{1}(X) = (L_{X})_{W_{X}}

The classification (but in fact only the part pertaining to finite Z_{p}-reflection groups (W, L) with Z_{p}[L]^{W} a polynomial algebra) also gives a verification that results of Borel, Steinberg, Demazure, and Notbohm [97, Prop. 1.11] extends to p-compact groups, p odd.

Theorem 1.8. Let X be a connected p-compact group, p odd. The following conditions are equivalent:

(1) X has torsion free Z_{p}-cohomology.

(2) Z_{p}[L]^{W} is a polynomial algebra over Z_{p}.

(3) All elementary abelian p-subgroups of X lie inside a maximal torus.

Even in the Lie case, the proof of the above theorem is still not entirely satisfactory despite much effort—see the comments surrounding our proof as well as Borel’s comments [8, p. 775] and the references [7, 36, 121]. Likewise the following related result from Lie theory is true.

Theorem 1.9. Let X be a connected p-compact group, p odd. Then the following conditions are equivalent:

(1) π_{1}(X) is torsion free.

(2) Every rank one elementary abelian p-subgroup ν : Z/p → X has connected centralizer C_{X}(ν) (= Ω map(BZ/p, BX)_{Bν} ; see Section 6).

(3) Every rank two elementary abelian p-subgroup is contained in a torus.

Results about p-compact groups can in general, via Sullivan’s arithmetic square, be translated into results about finite loop spaces, and the last theorem in this introduction is an example of such a translation. Recall that a finite loop space is a loop space (X, BX, e), where X is a finite CW-complex. A maximal torus of a finite loop space is simply a map BU(1)^{r} → BX for some r, such that the homotopy fiber is homotopy equivalent to a finite CW -complex of non-zero Euler characteristic. The classical maximal torus conjecture (stated in 1973 by Wilkerson [129, Conj. 1] as “a popular conjecture toward which the author is biased”), asserts that connected compact Lie groups are the only connected finite loop spaces which admit maximal tori. (A slightly more elaborate version states that the classifying space functor should set up a one to one correspondence between compact connected Lie groups, up to isomorphism, and connected finite loop spaces admitting a maximal torus, up to isomorphism, and furthermore the outer isomorphisms of G equal homotopy classes of self-homotopy equivalences of BG, the last part being true by [75].)

It is well known that a proof of the conjectured classification of p-compact groups for all primes p would imply the maximal torus conjecture. Our results at least imply that the conjecture is true after inverting the single prime 2.

Theorem 1.10. Let X be a connected finite loop space with a maximal torus. Then there exists a compact connected Lie group G such that BX[\frac{1}{2}] and BG[\frac{1}{2}] are homotopy equivalent spaces, where [\frac{1}{2}] indicates Z[\frac{1}{2}]-localization.

Relationship to the Lie case and the conjectural picture for p = 2. We now state a common formulation of both the classification of compact Lie groups and the classification of p-compact groups for p odd, which conjecturally should also hold for p = 2. We have not encountered this—in our opinion quite natural—description before in the literature (compare [41] and [80]).

Let R be an integral domain and W a finite R-reflection group. For an RW -lattice L (i.e., an RW -module which is finitely generated and free as an R-module) define SL to be the sublattice of L generated by (1 − w)x where w ∈ W and x ∈ L. Define an R-reflection datum to be a triple (W, L, L_{0}) where (W, L) is a finite R-reflection group and L_{0} is an RW - lattice such that SL ⊆ L_{0} ⊆ L and L_{0} is of the form SL′ for some other RW -lattice L′. (If R = Z_{p}, p odd, then “S” is idempotent (since H_{1}(W; L_{W}) = 0 for all (W, L) by a caseby-case computation given in [5, Thm. 3.2]) so L_{0} = SL in this case.) Two reflection data (W, L, L_{0}) and (W′, L′, L′_{0}) are said to be isomorphic if there exists an R-linear isomorphism φ : L → L′ such that φW φ^{−1} = W′ and φ(L_{0}) = L′_{0}.

If D is either the category of connected compact Lie groups or connected p-compact groups, then we can consider the assignment which to each object X in D assigns the triple (W, L, L_{0}), where W is the Weyl group, L = π_{1}(T) is the dual weight lattice, and L_{0} = ker(π_{1}(T) → π_{1}(X)) is the coroot lattice.

Theorem 1.1 and 1.7 as well as the classification of compact connected Lie groups (see e.g. [15]) can now be reformulated as follows:

Theorem 1.11. If D is the category of connected compact Lie groups or p-compact groups for p odd, then (W, L, L_{0}) is an R-reflection datum (R = Z for compact Lie groups and Z_{p} for p-compact groups) and this assignment sets up a bijection between the elements of D up to isomorphism and R-reflection datum, up to isomorphism. Furthermore the outer automorphisms of X equal outer automorphisms of the corresponding R-reflection datum.

Conjecture 1.12. Theorem 1.11 is also true if D is the category of connected 2-compact groups.

One can check that the conjecture on objects is equivalent to the conjecture given in [41] and [80], and the self-map statement would then follow from [75] and [98]. (The role of the coroot lattice L_{0} in the above theorem and conjecture is in fact in practice only to be able to distinguish direct summands isomorphic to SO(2n + 1) from direct summands isomorphic to Sp(n) (see Remark 7.3 and Proposition 7.4); alternatively on can use the extension class γ ∈ H^{3}(W; L_{X}) of the maximal torus normalizer (see Section 3) rather than L_{0} but in that picture it is not a priori clear which triples (W, L, γ) are realizable.) It would be desirable to give a “topological” version of Theorem 1.11 and Conjecture 1.12 which were statements on the level of automorphism spaces like Theorem 1.4 but we do not know a general formulation which incorporates this feature.

Organization of the paper and notation. The sections of this paper can be read in an almost arbitrary order. The short Section 2 sets up the map from space of automorphism of X to the space of automorphisms of N_{X}, and in Section 3 we give an algebraic description of the automorphism group of N_{X} (which we expand on in Section 13). In Section 4 we reduce the classification Theorem 1.1 to the case of simple, center-free, connected pcompact groups. In Section 5 we prove a theorem about invariant rings and show how this leads to an easy construction of the exotic p-compact groups. In Section 6 we give proofs of the main theorem, modulo obstruction group calculations which are carried out in Section 10. The applications listed in the introduction are proved in Section 7. The purely algebraic Section 8 contains complete information about all non-toral elementary abelian psubgroups of the exceptional Lie groups, along with their Weyl groups and centralizers, and Section 9 gives the analogous (but much easier) results for the projective unitary groups. (This information is used in a crucial way for the calculations in Section 10 as well as, in a milder way, directly in Section 6 for information about rank two non-toral subgroups.) In the appendix Section 11 we give a classification of finite Z_{p}-reflection groups generalizing Notbohm’s classification to all primes and in the appendix Section 12 we recall Notbohm’s results on invariant rings of Z_{p}-reflection groups. Finally in the appendix Section 13 we briefly calculate the outer automorphism group of the exotic finite Z_{p}-reflection groups to make the result of Theorem 1.1 more explicit.

We have tried to introduce all definitions relating to p-compact group as they are used, but it is probably nevertheless helpful for the reader unfamiliar with p-compact groups to keep copies of the excellent papers [48] and [49] of Dwyer-Wilkerson (whose terminology we follow) within reach. As a technical term we say that a p-compact group X is determined by N if it is true that any other p-compact group X′ with the same maximal torus normalizer is isomorphic isomorphic to X (which will be true for all p-compact groups, p odd, by Theorem 1.4).

Acknowledgments. We would like to thank H. H. Andersen, D. Benson, G. Kemper, A. Kleschev, G. Malle, and J.-P. Serre for helpful correspondence. We would in particular like to thank W. Dwyer, D. Notbohm, and C. Wilkerson for several useful tutorials on their beautiful work, which this paper builds upon.

2. The map Aut(BX) → Aut(BN_{X})

The purpose of this very short section is to construct the map Aut(BX) → Aut(BN_{X}) which we will later prove is an equivalence. We have been unable to find this description in the literature.

For a fibration f : E → B we let Aut(f) denote the space of commutative diagrams

\xymatrix{E \ar[d]^{f} \ar[r] & E \ar[d]^{f} \\ B \ar[r] & B}

such that the horizontal maps are homotopy equivalences. (This is a subspace of Aut(E) × Aut(B).)

Lemma 2.1 (Adams-Mahmud lifting). Let X be a p-compact group with maximal torus normalizer N_{X}. Turn the inclusion of the maximal torus normalizer into a fibration f : BN_{X} → BX. Then the restriction map Aut(f) → Aut(BX) is an equivalence of topological monoids.

In particular any self-homotopy equivalence of BX lifts to a self-homotopy equivalence of BN_{X}, which is unique in the strong sense that the space of lifts is contractible. Choosing an inverse to the equivalence Aut(f) → Aut(BX), we get by restriction a canonical homomorphism of topological monoids

Aut(BX) →^{≅} Aut(f) → Aut(BN_{X}).

Proof. For any φ ∈ Aut(BX), there exists, e.g. by [90, Thm. 1.2(3)], a ψ ∈ Aut(BN_{X}) such that φf is homotopic to fψ. Since f is assumed to be a fibration, ψ can furthermore be modified such that the equality is strict. This shows that the evaluation map Aut(f) → Aut(BX) is surjective on components. This map of group-like topological monoids is furthermore easily seen to have the homotopy lifting property. To see that it is a homotopy equivalence we hence just have to verify that the fiber over the identity map Aut_{BX}(BN_{X}) is contractible. We have the following diagram with rows and columns fibrations

[figure: W_{X}; X/T; X/N_{X}; W_{X}; BT; BN_{X}; ∗; BX; BX.]

Taking homotopy T -fixed points of the top row produces a fibration sequence W_{X} → (X/T)^{hT} → (X/N_{X})^{hT}, where, by the definition of W_{X}, the inclusion of the fiber in the total space is a homotopy equivalence. (We refer to [48, 3.3, §10] for basic facts and definitions about homotopy actions.) Hence (X/N_{X})^{hT} is contractible. But then (X/N_{X})^{hN_{X}} is contractible as well, since (X/N_{X})^{hN̆_{X}} ≃ ((X/N_{X})^{hT̆})^{hW_{X}}, where N̆_{X} is a discrete approximation to N_{X} (cf. [48, Prop. 3.13]). Since the space of maps BN_{X} → BN_{X} over BX identifies with (X/N_{X})^{hN_{X}}, we see that any self-map of BN_{X} over BX is an equivalence, and that Aut_{BX}(BN_{X}) is contractible as wanted. □

3. Automorphisms of maximal torus normalizers

The purpose of this short section is establish some easy facts about automorphisms of maximal torus normalizers which are needed to carry out the reduction to connected, centerfree simple p-compact groups in Section 4. At the same time the section serves to make the automorphism statement of Theorem 1.1 more explicit.

Proposition 3.1. Suppose that N is an extended p-compact torus, i.e., a loop space such that W = π_{0}(N) is a finite group and the identity component of N is a p-compact torus T. Let N̆ be the unique discrete approximation to N (see [48, 3.12]), and recall that N̆ will have a unique largest p-divisible subgroup T̆, which will be the discrete approximation of T.

The obvious map associating to a self-homotopy equivalence of BN̆ a self-homotopy equivalence of BN (via fiberwise F_{p}-completion; cf. [17, Ch. I §8]) induces a homotopy equivalence of aspherical group-like topological monoids

Aut(BN̆)_{p}^{∧} →^{≅} Aut(BN).

If π_{0}(N) acts faithfully on π_{1}(N_{1}) then Aut_{1}(BN̆), the component of Aut(BN̆) of the identity map, has the homotopy type of B(T̆^{W}) where T̆ is a discrete approximation to T.

Sketch of proof. The statement on the level of component groups follows directly from [48, 3.12]. (The point is that the homotopy fiber of BN̆ → BN will have homotopy type K(V, 1) for a Q_{p} vector space V, and hence the existence and uniqueness to lifting a map BN̆ → BN̆ to BN lie in H^{n}(N̆; V) where n = 2, 1 which are easily seen to be zero.) It is likewise easy to see that both spaces are aspherical and that we get a homotopy equivalence on component groups. The last statement is likewise obvious. □

Let L be a finitely generated free Z_{p}-module and suppose that W ⊆ GL(T̆), where we set T̆ = L⊗Z/p^{∞}. Consider the second cohomology group H^{2}(W; T̆) which classifies extensions of W by T̆ with the fixed action of W on T̆. Given an isomorphism α : L → L′ sending W ⊆ GL(L) to W′ ⊆ GL(L′) we get an isomorphism of cohomology groups H^{2}(W; T̆) →

H^{2}(W′; T̆′) by sending an extension T̆ →^{i} N̆ →^{p} W to the extension T̆′ →^{i∘α^{−1}} N̆ →^{c_{α}∘p} W′, where c_{α} denotes conjugation by α. An isomorphism between two triples (W, L, γ) and (W′, L′, γ′) is an isomorphism L → L′ sending W to W′ and γ to γ′. The automorphism group of a triple (W, L, γ) thus identifies with

γN_{GL(T̆)}(W) = {α ∈ N_{GL(T̆)}(W) | α(γ) = γ ∈ H^{2}(W; T̆)}.

It follows directly from the definition (and using that T̆ is characteristic in N̆) that two triples as above are isomorphic if and only if the associated groups N̆ and N̆′ are isomorphic, where N̆ is obtained from the extension 1 → T̆ → N̆ → W → 1 given by γ, and analogously for γ′. However, N̆ and (W, L, γ) in general have slightly different automorphisms, as is described in the following lemma (see also [128]):

Proposition 3.2. In the notation above, for any exact sequence 1 → T̆ → N̆ →^{π} W → 1 with extension class γ we have a canonical exact sequence

(3.1)  1 → Der(W, T̆) → Aut(N̆) → ^{γ}N_{GL(T̆)}(W) → 1

where we embed the derivations Der(W, T̆) in Aut(N̆) by sending a derivation s to the automorphism given by x ↦ s(π(x))x, and the map Aut(N̆) → ^{γ}N_{GL(T̆)}(W) is given by

restricting an automorphism φ ∈ Aut(N̆) to T̆.

This exact sequence has a exact subsequence 1 → T̆/T̆^{W} → N̆ /ZN̆ → W → 1 and the quotient exact sequence is

1 → H^{1}(W, T̆) → Out(N̆) → ^{γ}N_{GL(T̆)}(W)/W → 1.

In particular if (W, L) is a finite Z_{p}-reflection group and p is odd then by [5],[74, proof of Prop. 3.5] H^{1}(W; T̆) = 0, so we get an isomorphism Out(N̆) →^{≅} ^{γ}N_{GL(T̆)}(W)/W.

Proof. Let φ ∈ Aut(N̆), and consider the restriction map φ ↦ φ_{|T̆} ∈ Aut(T̆). Note that

for all x ∈ N̆, l ∈ T̆ we have

(φ ∘ c_{x})(l) = φ(xlx^{−1}) = φ(x)φ(l)φ(x)^{−1} = (c_{φ(x)} ∘ φ)(l)

so φ_{|T̆} ∈ N_{GL(T̆)}(W). That the image is the elements which fixes the extension class follows easily from the definitions: The diagram

[figure: T̆; i∘φ; N̆; c_{φ}∘π; φ; W; T̆; i; N̆; π; W]

shows that φ leaves γ invariant. Likewise, to see that the right map in (3.1) is surjective let ψ ∈ ^{γ}N_{GL(T̆)}(W) and let T̆ → Ñ → W be the extension obtained by first pushing forward along ψ : T̆ → T̆ and then pulling back along ψ^{−1}(−)ψ : W → W. Since ψ fixes γ there exists an isomorphism Ñ → N̆ making the following diagram commute

[figure: T̆; ψ; T̆; T̆; N̆; Ñ; N̆; W; ψ(−)ψ^{−1}; W; W]

which shows that Aut(N̆) → ^{γ}N_{GL(T̆)}(W) is surjective.

Now suppose φ ∈ Aut(N̆) restricts to the identity on T̆. For a given x ∈ N̆ we have

φ(x)lφ(x^{−1}) = φ(x)φ(l)φ(x^{−1}) = φ(xlx^{−1}) = xlx^{−1}

so the induced map φ : W → W is the identity since W acts faithfully on T̆. This means that we can define a map s : W → T̆ by s(w) = w̃^{−1}φ(w̃) where w̃ is a lift of w, and this is easily seen to be a derivation. Furthermore taking the automorphism of N̆ associated to s gives back φ, which establishes exactness in the middle, and we have proven that we have the first exact sequence.

The existence of the short exact subsequence is clear, noting that ZN̆ = T̆^{W} (since W acts faithfully on T̆) and that T̆/T̆^{W} embeds in Der(W; T̆) as the principal derivations by sending l to the derivation w ↦ c_{w}(l)l^{−1}. The last exact sequence is now obvious. □

Remark 3.3. See [66] for a related exact sequence for compact Lie groups, fitting with the conjectured classification of p-compact groups for p = 2.

Proposition 3.4. Suppose {(W_{i}, L_{i}, γ_{i})}_{i=1}^{k} is a collection of non-isomorphic triples where L_{i} is a finitely generated free Z_{p}-module, W_{i} is a finite subgroup of GL(L_{i}) such that L_{i} ⊗Q is irreducible, and γ_{i} ∈ H^{2}(W_{i}; T̆_{i}). Let (W, L, γ) = ×_{i=1}^{k}(W_{i}, L_{i}, γ_{i})^{m_{i}} denote the product. Then

×_{i=1}^{k}((^{γ_{i}}N_{GL(L_{i})}(W_{i})/W_{i}) ≀ Σ_{m_{i}}) →^{≅} ^{γ}N_{GL(L)}(W)/W.

Proof. Assume for ease of notation that L = L_{1} ⊕ L_{2}; the general case follows from this by induction. Consider φ ∈ N_{GL(L_{1}⊕L_{2})}(W_{1} × W_{2}). For every w ∈ W_{1} × W_{2} there exists a unique w̃ ∈ W_{1} × W_{2} such that

φ(wx) = w̃φ(x) for all x ∈ L.

Let α denote the element in Aut(W_{1} × W_{2}) given by w ↦ w̃.

By the definition of α the canonical map

φ_{ji} : L_{i} → L_{1} ⊕ L_{2} →^{φ} ^{α}(L_{1} ⊕ L_{2}) → ^{α}L_{j}

is W_{1} ×W_{2}-equivariant, where the superscript α means that we are acting through α. Hence this map, after tensoring with Q has to be either an isomorphism or zero, since L_{i} ⊗ Q and αL_{j} ⊗ Q are irreducible. But combined with the fact that φ is an isomorphism, this means that

φ = ( \begin{matrix}φ_{11} & φ_{12} \\ φ_{21} & φ_{22} \end{matrix} )

has to consist of either just “diagonal” elements or just “off diagonal” elements. If we furthermore require φ to respect the extension classes it is clear that we can only have the “diagonal” case if (W_{1}, L_{1}, γ_{1}) is not isomorphic to (W_{2}, L_{2}, γ_{2}) whereas if they are isomorphic both cases can really occur. This proves the lemma. □

4. Reduction to connected, center-free simple p-compact groups

In this section we prove some lemmas, which, together with the splitting theorems of Dwyer-Wilkerson [50] and Notbohm [99], reduce the proof of Theorem 1.4 to the case of connected, center-free simple p-compact groups. (Recall that a connected p-compact group is called simple if (W_{X}, L_{X} ⊗Q) is an irreducible Q_{p}-reflection group; it is called center-free if Aut_{1}(BX) is contractible—equivalent definitions are given in [49].) This reduction is known and most of it appears in [89] (relying on earlier work of that author). We here provide a self-contained and a bit more direct proof using [49].

Lemma 4.1 (Behavior with respect to products). Let X and X′ be p-compact groups with maximal torus normalizers N and N′. Then N × N′ is a maximal torus normalizer for X × X′ and the following statements hold:

(1) Aut_{1}(BX) × Aut_{1}(BX′) →^{≅} Aut_{1}(BX × BX′) and Aut_{1}(BN) × Aut_{1}(BN′) →^{≅} Aut_{1}(BN × BN′), where Aut_{1} denotes homotopy equivalences homotopic to the identity.

(2) If Aut(BX) → Aut(BN) and Aut(BX′) → Aut(BN′) are injective on π_{0}, then so is Aut(B(X × X′)) → Aut(B(N × N′)).

(3) Suppose p is odd and that X and X′ are connected and center-free. If Aut(BX) → Aut(BN) and Aut(BX) → Aut(BN′) are surjective on π_{0}, then so is Aut(B(X × X′)) → Aut(B(N × N′)).

Proof. Recall that the map Aut(BX) → Aut(BN) was described in Section 2. To see (1) first note that (4.1) map(BX × BX′, BX × BX′) ≃ map(BX, map(BX′, BX)) × map(BX′, map(BX, BX′)).

The evaluation map map(BX′, BX)_{const} → BX is an equivalence by the Sullivan conjecture for p-compact groups [49, Thm 9.3 and Prop. 10.1], (and likewise with X and X′ switched). Since the component of the identity map on the left hand side of (4.1) lands in the component of the constant map in map(BX′, BX) this shows that map(BX × BX′, BX × BX′)_{1} ≃ map(BX, BX)_{1} × map(BX′, BX′)_{1} as wanted. (The statement just says that the center of a product of p-compact groups is the product of the centers, which of course also follows from the equivalence of the different definitions of the center from [49].)

To see (2) suppose that φ is a self-equivalence of BX × BX′ such that its restriction to a self-equivalence of B(N × N′) becomes homotopic to the identity. The restriction φ_{|BX×∗} composed with the projection onto BX′ becomes null homotopic upon restriction to BN, which by e.g, [91, Thm. 6.1] implies that it is null homotopic. Likewise the projection of φ_{|∗×BX′} onto BX′ becomes homotopic to the identity map upon restriction to BN, which by assumption means that the projection of φ_{∗×BX′} onto BX′ is the identity. But by adjointness, repeating the argument of the first claim, this implies that φ composed with the projection onto BX′ is homotopic to the projection map onto BX′ (this is [49, Lem. 5.3]). By symmetry this holds for the projection onto BX as well, and we conclude that φ is homotopic to the identity as wanted.

Finally, combining Proposition 3.2 and 3.4 gives (3), since π_{0}(BN) = Out(N̆) by Proposition 3.1. □

Remark 4.2. Claim (3) of the above lemma is in general false if p = 2. For instance if X = SO(3)_{2}^{∧} then both for Y = X and Y = X × X we have π_{0}(Aut(BY)) ≅ N_{GL(L_{Y} )}(W_{Y})/W_{Y}. But for Y = X × X we have H^{1}(W_{Y}; T̆_{Y}) ≅ Z/2 × Z/2, so BN_{Y} (T) has non-trivial automorphisms which restrict to the identity on BT (see Proposition 3.2).

Recall the observation that for p odd the component group of X is determined by W_{X}:

Lemma 4.3. Let X be a p-compact group for p odd, with maximal torus normalizer j : N → X. By definition W_{X} = π_{0}(N). The map π_{0}(j) : W_{X} = π_{0}(N) → π_{0}(X) is surjective. The kernel is O^{p}(W_{X}), the subgroup generated by elements of order prime to p. It can also be identified with the Weyl group of the connected component X_{1} of X, and is the largest Z_{p}-reflection subgroup of W_{X}.

Proof. By [49, Rem. 2.11] π_{0}(j) is surjective with kernel the Weyl group of the connected component of X. Since π_{0}(X) is a p-group, O^{p}(π_{0}(N)) is contained in the kernel. On the other hand, since p is odd, the Weyl group of X_{1} is generated by elements of order prime to p, since it is a Z_{p}-reflection group, so equality has to hold. □

Remark 4.4. For p = 2 the component group of X cannot be read off from N_{X}, and one would have to remember π_{0}(X) as part of the data. The 2-compact groups SO(3)_{2}^{∧} and O(2)_{2}^{∧} have for example the same maximal torus normalizer, namely O(2)_{2}^{∧}. (Note however that if X is the centralizer of a toral abelian subgroup A of a connected p-compact group Y, then the component group of X can be read off from A and N_{Y} (see [49, Thm. 7.6]); a case of frequent interest.)

Before proceeding recall that by [43] (see also [49, Prop. 11.9]) we have, for a fibration

F → E →^{f} B, a fibration sequence

map(B, B Aut(F))_{C(f)} → B Aut(f) → B Aut(B).

Here C(f) denotes the components corresponding to the orbit of the π_{0}(Aut(B))-action on the class in [B, B Aut(F)] classifying the fibration.

We are interested in when the map of monoids Aut(f) → Aut(E) is a homotopy equivalence. This will follow if we can see that Aut_{1}(f) → Aut_{1}(E) and π_{0}(Aut(f)) → π_{0}(Aut(E)) are equivalences. By an easy general argument given as [49, Prop. 11.10] the statement about identity components follows if B → map(F, B)_{0} is an equivalence.

Lemma 4.5 (Behavior with respect to components). Let X be a p-compact group with maximal torus normalizer N, and assume that p is odd (so that π_{0}(X) can be read off from N). Let N_{1} denote the kernel of the map N → π_{0}(X), and note that N_{1} is a maximal torus normalizer for X_{1}.

If Aut(BX_{1}) →^{≅} Aut(BN_{1}), then Aut(BX) →^{≅} Aut(BN). If furthermore BX_{1} is determined by BN_{1} then BX is determined by BN.

Proof. First note that by an inspection of Euler characteristics and using [90, Thm. 1.2(3)], N_{1} is really a maximal torus normalizer in X_{1} as claimed. Set π = π_{0}(X) for short. We want to apply the setup described before the lemma to the fibrations BX_{1} → BX → Bπ and BN_{1} → BN → Bπ and to see that in both cases the map of monoids Aut(f) → Aut(E) are homotopy equivalences. By the remarks above this follows if it is an isomorphism on π_{0} and that B → map(F, B)_{0} is an equivalence. The statement about π_{0} is true in both cases since a self-map of E determines a unique self-map of Bπ. Likewise it is easy to see that Bπ →^{≃} map(BX_{1}, Bπ)_{0} and that Bπ →^{≃} map(BN_{1}, Bπ)_{0}. This means that our map B Aut(BX) → B Aut(BN) (from Lemma 2.1) fits in a map of fibration sequences

\xymatrix{map(Bπ, B Aut(BX_{1}))_{C(f)} \ar[d] \ar[r] & B Aut(BX) \ar[d] \ar[r] & B Aut(Bπ) \ar[d] \\ map(Bπ, B Aut(BN_{1}))_{C(f)} \ar[r] & B Aut(BN) \ar[r] & B Aut(Bπ).}

Here the maps between the fibers and bases are homotopy equivalences by assumption, and we conclude that we get homotopy equivalences between the total spaces as well.

Now assume furthermore that X_{1} is determined by N_{1}, and let X′ be another p-compact group with maximal torus normalizer N. By Lemma 4.3 we get that π = π_{0}(X) ≅ π_{0}(X′) and that N_{1} is also a maximal torus normalizer in X′_{1}.

We want to show that the two fibrations BX → Bπ and BX′ → Bπ are equivalent as fibrations over Bπ, or equivalently that the π-spaces BX_{1} and BX′_{1} are hπ-equivalent, i.e., that we can find a zig-zag of π-maps which are non-equivariant equivalences connecting the two (see e.g., [38] where this equivalence relation is called equivariant weak homotopy equivalence).

By the assumptions on X_{1} we can choose a homotopy equivalence Bf : BX_{1} → BX′_{1} such that

\xymatrix{ & BN_{1} \ar[dr]^{Bj′} \ar[dl]_{Bj} \\ BX_{1} \ar[rr]^{Bf} &  & BX′_{1}}

commutes up to homotopy, and Bf is unique up to homotopy.

We now want to see that we can change Bf so that it becomes a π-map. For this, consider the restriction π-map

map(BX_{1}, BX′_{1}) → map(BN_{1}, BX′_{1}).

By the assumptions on Aut(BX_{1}) this map sends distinct components of map(BX_{1}, BX′_{1}) corresponding to homotopy equivalences to distinct components of map(BN_{1}, BX′_{1}). Moreover, by the proof of Lemma 2.1, we have a homotopy equivalence map(BX_{1}, BX′_{1})_{Bf} ≃ map(BN_{1}, BX′_{1})_{Bf∘Bj}. In particular the component map(BX_{1}, BX′_{1})_{Bf} is preserved under the π-action, since this obviously is so for map(BN_{1}, BX′_{1})_{Bj′}. Furthermore since map(BN_{1}, BX′_{1})_{Bj′}^{π} contains Bj′ we see that map(BX_{1}, BX′_{1})_{Bf}^{hπ} ≃ map(BN_{1}, BX′_{1})_{Bj′}^{hπ} is non-empty, and so there exists a π-map Eπ×BX_{1} → BX′_{1} which is a homotopy equivalence. This shows that BX_{1} and BX′_{1} are hπ-homotopy equivalent as wanted. □

Remark 4.6. If X is a connected p-compact group, and p is odd, then it follows from [49, Thm. 7.5] that Z(N̆) is a discrete approximation to the center of X. The proof of the above lemma extends this to X non-connected provided we know that the self-maps of X_{1} are detected by their restriction to N_{1}, which will be a consequence of Theorem 1.4. Having to appeal to this is a bit unfortunate but seems unavoidable. The point is that if there for a connected p-compact group X, existed a self-equivalence σ of X of finite p-power order and not detected by N, then we could form X⋊⟨σ⟩, where σ would be central in the normalizer but not in the whole group. (See also Lemma 10.2.)

Lemma 4.7 (Behavior with respect to centers). Let X be a connected p-compact group with center Z.

(1) If π_{0}(Aut(BX/Z)) → π_{0}(Aut(BN /Z)) is surjective and X/Z is determined by N /Z then X is determined by N.

(2) If p is odd and Aut(BX/Z) → Aut(BN /Z) is a homotopy equivalence then Aut(BX) → Aut(BN) is as well.

Proof. Suppose that X and X′ have the same maximal torus normalizer N and choose fixed inclusions j : N → X and j′ : N → X′. By [49, Thm. 7.6] X′ and X have the same center Z. Suppose that X/Z is isomorphic to X′/Z. If π_{0}(Aut(BX/Z)) → π_{0}(Aut(BN /Z)) is surjective we furthermore have that we can choose the homotopy equivalence BX/Z → BX/Z in such a way that

\xymatrix{ & BN /Z \ar[dl]_{j/Z} \ar[dr]^{j′/Z} \\ BX/Z \ar[rr] &  & BX′/Z}

commutes up to homotopy.

We have canonical maps BX → B^{2}Z and BX′ → B^{2}Z classifying the extensions, and we claim that in fact the bottom triangle in the diagram

\xymatrix{ & BN /Z \ar[dr] \ar[dl] \\ BX/Z \ar[dr] \ar[rr] &  & BX′/Z \ar[dl] \\  & B^{2}Z}

commutes up to homotopy. By construction the outer square commutes up to homotopy (since both composites agree with the map classifying map BN /Z → B^{2}Z since we kept j and j′ fixed). Since the top triangle also commutes up to homotopy, an application of the transfer [48, 9.12], using that B^{2}Z is a product of Eilenberg-Mac Lane spaces and that χ((X/Z)/(N /Z)) = 1, shows that the bottom triangle commutes up to homotopy as well. Since we have constructed a map BX/Z → BX′/Z over B^{2}Z we get an induced homotopy equivalence BX → BX′. (Note that this construction does not a priori give this map as a map under BN.)

We now want to get the second statement about automorphism groups. Consider the homotopy commutative diagram

\xymatrix{BN \ar[r]^{f′} \ar[d] & BN /Z \ar[d] \\ BX \ar[r]^{f} & BX/Z}

where we can suppose that the two horizontal maps f′ and f are fibrations.

We first claim that we can replace B Aut(f) with B Aut(BX) and B Aut(f′) with B Aut(BN). As in the case of the component group (see the proof of Lemma 4.5) we just have to justify that in the appropriate fibration sequences we have equivalences B → map(F, B)_{0} and π_{0}(Aut(f)) → π_{0}(Aut(E)). The map BX/Z → map(BZ, BX/Z)_{0} is a homotopy equivalence since the trivial map is central [49, Prop. 10.1]. That BN /Z → map(BZ, BN /Z)_{0} is an equivalence is a similar (but easier) argument.

By Lemma 2.1 a self-equivalence of BX induces a unique self-equivalence of BN, and hence a canonical self-equivalence of BZ. Now, by the description of X/Z as a Borel construction (given in [48, proof of Prop. 8.3]) we get a canonical self-equivalence of BX/Z. This self-equivalence is furthermore unique, in the sense that given a diagram

\xymatrix{BX \ar[r]^{g} \ar[d] & BX \ar[d] \\ BX/Z \ar[r]^{g′} & BX/Z}

the homotopy type of g′ is uniquely given by that of g. To see this note that by Lemma 2.1 the diagram restricts to a unique diagram

\xymatrix{BN \ar[r]^{g̃} \ar[d] & BN \ar[d] \\ BN /Z \ar[r]^{g̃′} & BN /Z.}

By looking at discrete approximations we see that the homotopy class of g̃′ is determined by g̃. Since by assumption the homotopy class of g′ is determined by g̃′, we conclude that a self-equivalence of BX induces a unique self-equivalence of BX/Z, and so π_{0}(Aut(f)) ≅ π_{0}(Aut(BX)). The last part of the argument furthermore shows that also π_{0}(Aut(f′)) ≅ π_{0}(Aut(BN)).

We hence have the following diagram where horizontal maps are fibration sequences

\xymatrix{map(BX/Z, B Aut(BZ))_{C(f)} \ar[r] \ar[d] & B Aut(BX) \ar[r] \ar[d] & B Aut(BX/Z) \ar[d] \\ map(BN /Z, B Aut(BZ))_{C(f′)} \ar[r] & B Aut(BN) \ar[r] & B Aut(BN /Z)).}

Examining when the middle vertical arrow is a homotopy equivalence reduces to finding out when the restriction map map(BX/Z, B Aut(BZ))_{C(f)} → map(BN /Z, B Aut(BZ))_{C(f′)} is a homotopy equivalence, which we now proceed to analyze.

Note that since BZ is an Eilenberg-Mac Lane space we have a fibration sequence

B^{2}Z → B Aut(BZ) → B Aut(Z̆)

where Z̆ is the discrete approximation to Z and Aut(Z̆) is the discrete group of automorphisms. Since our extensions are central this gives a diagram of fibration sequences

\xymatrix{map(BX/Z, B^{2}Z)_{C(f)} \ar[r] \ar[d] & map(BX/Z, B Aut(BZ))_{C(f)} \ar[r] \ar[d] & map(BX/Z, B Aut(Z̆))_{0} \ar[d] \\ map(BN /Z, B^{2}Z)_{C(f′)} \ar[r] & map(BN /Z, B Aut(BZ))_{C(f′)} \ar[r] & map(BN /Z, B Aut(Z̆))_{0}.}

Again, in this diagram the map between the base spaces is obviously an equivalence, so we are reduced to studying

(4.2)  map(BX/Z, B^{2}Z)_{C(f)} → map(BN /Z, B^{2}Z)_{C(f′)}.

Since B^{2}Z is a product of Eilenberg-Mac Lane spaces a transfer argument (cf. [48, 9.12]) shows that this gives an embedding as a retract. Since we assume π_{0}(Aut(BX/Z)) ≅ π_{0}(B Aut(BN /Z)) we furthermore get that this is an isomorphism on π_{0} by the definition of C(f) and C(f′). Now set X′ = X/Z and N′ = N /Z and let (W, L′) denote the Weyl group of X′. Write BZ ≃ B^{2}A × BA′, where A is torsion free and A′ is finite (see [49, Thm. 1.1]). On π_{1} the map (4.2) identifies with

H^{1}(BX′; A′) ⊕ H^{2}(BX′; A) → H^{1}(BN′; A′) ⊕ H^{2}(BN′; A).

The group H^{1}(BN′; A′) is zero since π_{1}(BN′) = W_{X} is generated by elements of order prime to p, since p is assumed to be odd.

Furthermore, H^{2}(BN; A) is related via the Serre spectral sequence to the groups

H^{2}(BW; H^{0}(B^{2}L′; A)), H^{1}(BW; H^{1}(B^{2}L′; A)), and H^{0}(BW; H^{2}(B^{2}L′; A)).

The first of these groups is zero since W is generated by elements prime to p by the assumption that p is odd. The second is obviously zero, and the last group is zero since H^{0}(W; Hom(L′, Z_{p})) = Hom((L′)_{W} , Z_{p}) = 0 because (L′)_{W} is finite.

Hence we get an isomorphism on π_{1}, since we already know that the map is injective. On π_{2} and π_{3} the map identifies with H^{0}(BX′; A′)⊕H^{1}(BX′; A) → H^{0}(BN′; A′)⊕H^{1}(BN′; A) and H^{0}(BX′; A) → H^{0}(BN′; A) respectively, and these maps are obviously isomorphisms. Hence map(BX/Z, B^{2}Z)_{C(f)} → map(BN /Z, B^{2}Z)_{C(f′)} is a homotopy equivalence, which via the fibration sequences above imply that B Aut(BX) → B Aut(BN) is a homotopy equivalence as wanted. □

Remark 4.8. Consider BX = B(SO(3) × S^{1})_{2}^{∧}. This has center Z = S^{1}_{2}^{∧} and X/Z = SO(3)_{2}^{∧}. It is easy to calculate directly that B Aut(BX/Z) →^{≃} B Aut(BN /Z) (or appeal to [75]). However π_{0}(Aut(BX)) → π_{0}(Aut(BN)) is not onto by Proposition 3.2, since Hom(W_{SO(3)}, Z/2^{∞}) = Z/2. This shows that the p odd assumption is necessary in the last part of the above lemma. (Compare also Remark 4.2.)

Remark 4.9. Suppose that X is a connected p-compact group. Fibration sequences with base B^{2}π_{1}(X) and fiber B(X⟨1⟩) are one-to-one correspondence with the set of maps [B^{2}π_{1}(X), B Aut(B(X⟨1⟩))]. Likewise self-equivalences of BX can be expressed in terms of self-equivalences of B(X⟨1⟩) and π_{1}(X), analogously to the lemmas above. Hence if we a priori knew that Theorem 1.7 held true, i.e., if we could read off π_{1}(X) from N_{X} then the above methods would reduce the proof of the main theorems to the simply connected case, which could be used advantageously in the proofs. (See also Remark 7.3.)

Remark 4.10. The assumption in Lemma 4.7(1) that π_{0}(Aut(BX/Z)) → π_{0}(Aut(BN /Z)) is surjective has the following origin. We have a canonical restriction map H^{2}(BX/Z; Z̆) → H^{2}(BN /Z; Z̆), which is injective by a transfer argument. Two extension classes in H^{2}(BX/Z; Z̆) give arise to isomorphic total spaces if the extension classes are conjugate via the actions of Aut(BX/Z) and Aut(Z̆) on this H^{2}. The total spaces have isomorphic maximal torus normalizer if the extension classes have images in H^{2}(BN /Z; Z̆) which are conjugate under the action of Aut(BN /Z) and Aut(Z̆), which could a priori be a weaker notion.

5. An integral version of a theorem of Nakajima and realization of p-compact groups

The goal of this section is to prove an integral version of an algebraic result of Nakajima (Theorem 5.1) and use this to prove a theorem (Theorem 5.3) which, as part of our induction proof of Theorem 1.1, will allow us to construct the center-free p-compact groups corresponding to Z_{p}-reflection groups (W, L) such that Z_{p}[L]^{W} is a polynomial algebra. This will provide the existence part of Theorem 1.1. We feel that this way of showing existence, is perhaps more straightforward that previous approaches; compare for instance [97]. (We refer to the introduction for the history behind this result.)

Theorem 5.1. Let p be an odd prime and let (W, L) be a finite Z_{p}-reflection group. For a subgroup V of L ⊗ F_{p} we let W_{V} denote the pointwise stabilizer of V in W. Then the following conditions are equivalent:

(1) Z_{p}[L]^{W} is a polynomial algebra.

(2) Z_{p}[L]^{W_{V}} is a polynomial algebra for all non-trivial subspaces V ⊆ L ⊗ F_{p}.

(3) (W_{V} , L) is a Z_{p}-reflection group for all non-trivial subspaces V ⊆ L ⊗ F_{p}.

Remark 5.2. An analog of the implication (1) ⇒ (2) where the ring Z_{p} is replaced by a field was proven by Nakajima [92, Lem. 1.4] (in the case of finite fields see also [53] and [94]). For fields of positive characteristic the implication (3) ⇒ (1) does not hold; see [78] for more information about this case. Our proof unfortunately involves the classification of finite Z_{p}-reflection groups and some case-by-case checking. (See the discussion following the proof of Theorem 1.8 for related information.)

Proof of Theorem 5.1. To start, note that the implication (2) ⇒ (3) follows from the fact that if Z_{p}[L]^{W_{V}} is a polynomial algebra then Q_{p}[L ⊗ Q]^{W_{V}} is as well, so (W_{V} , L) is a Z_{p}-reflection group by the Shephard-Todd-Chevalley theorem ([6, Thm. 7.2.1] or [116, Thm. 7.4.1]).

To go further we want to see that the theorem is well behaved under products, i.e., that if (W, L) = (W′, L′) × (W′′, L′′), then the theorem holds for (W, L) if it holds for (W′, L′) and (W′′, L′′). This follow from the fact that the stabilizer in W′ ×W′′ of an arbitrary subgroup in (L′ ⊗ F_{p}) ⊕ (L′′ ⊗ F_{p}) equals the stabilizer of the smallest product subgroup containing it, combined with the fact that the tensor product of two algebras is a polynomial algebra if and only if each of the factors are. Hence to prove the remaining implications it follows from Theorem 11.1 that it suffices to consider separately the cases where (W, L) comes from a compact connected Lie group and the cases where (W, L) is one of the exotic Z_{p}-reflection groups.

Assume first that (W, L) = (W_{G}, L_{G} ⊗ Z_{p}) for a compact connected Lie group G. If Z_{p}[L]^{W} is a polynomial algebra then by Theorem 12.2 (which involves case-by-case considerations and p odd) BX = BG_{p}^{∧} has the property H^{∗}(BX; Z_{p}) ≅ H^{∗}(B^{2}L; Z_{p})^{W}. We can identify V ⊆ L ⊗ F_{p} with a toral elementary abelian p-subgroup in X and by [53, Rem. 1.3] H^{∗}(BC_{X}(V); Z_{p}) is again a polynomial algebra concentrated in even degrees. In particular C_{X}(V) is connected and by [49, Thm. 7.6] W_{C_{X}(V)} = W_{V}. Hence, by Theorem 12.1, H^{∗}(BC_{X}(V); Z_{p}) ≅ H^{∗}(B^{2}L; Z_{p})^{W_{V}}, so Z_{p}[L]^{W_{V}} is a polynomial algebra. This shows that (1) ⇒ (2) when (W, L) comes from a compact Lie group. To prove (3) ⇒ (1) for Lie groups suppose that (W, L) is a finite Z_{p}-reflection group corresponding to a p-compact group X = G_{p}^{∧} such that (W_{V} , L) is a Z_{p}-reflection group for all non-trivial V ⊆ L ⊗ F_{p}. Since p is odd it follows by [49, Thm. 7.6] that C_{X}(V) is connected for all non-trivial V ⊆ L ⊗ F_{p}. Hence, since X is assumed to come from a compact Lie group [7, Thm. B] (or [121, Thm. 2.28]) implies that H^{∗}(BX; Z_{p}) does not have p-torsion and hence H^{∗}(BX; Z_{p}) ≅ H^{∗}(B^{2}L; Z_{p})^{W} (cf. Theorem 12.1). So Z_{p}[L]^{W} is a polynomial algebra as wanted.

Next we assume that (W, L) is one of the exotic Z_{p}-reflection groups. By Theorem 12.2, Z_{p}[L]^{W} is a polynomial algebra, so we only need to prove that Z_{p}[L]^{W_{V}} is a polynomial algebra for any non-trivial V ⊆ L ⊗ F_{p}. Furthermore, by Theorem 12.2(2), F_{p}[L ⊗ F_{p}]^{W} is a polynomial algebra. Nakajima’s result [92, Lem. 1.4] shows that F_{p}[L ⊗ F_{p}]^{W_{V}} is a polynomial algebra as well. Thus we are done if p ∤ |W_{V} | by Lemma 12.6, which in particular covers the cases where p ∤ |W |.

If (W, L) belongs to family number 2 on the Clark-Ewing list, then since p is odd, it is easily seen from the form of the representing matrices (see Section 11 for a concrete description) that reduction mod p gives a bijection between reflections in (W, L) and (W, L⊗ F_{p}). As F_{p}[L ⊗ F_{p}]^{W_{V}} is a polynomial algebra it follows by the Shephard-Todd-Chevalley theorem [6, Thm. 7.2.1] that W_{V} ⊆ GL(L⊗F_{p}) is a reflection group. Thus (W_{V} , L) is a Z_{p}- reflection group. Since the representing matrices are monomial, it follows by [92, Thm. 2.4] that Z_{p}[L]^{W_{V}} is a polynomial algebra.

By Theorem 11.1 only four cases remain, namely the Zabrodsky-Aguadé cases (W_{12}, p = 3), (W_{29}, p = 5), (W_{31}, p = 5) and (W_{34}, p = 7). For each of these a direct computation (for instance easily done with the aid of a computer) shows that if S is a Sylow p-subgroup of W, then U = (L ⊗ F_{p})^{S} is 1-dimensional and (W_{U}, L) is equivalent to (Σ_{p}, L_{SU(p)} ⊗ Z_{p}) (the construction of such a subgroup can also be found in Aguadé [3]). Hence we see that if V ⊆ L ⊗ F_{p} is non-trivial then either p ∤ |W_{V} | or V is W -conjugate to U. But in these cases we already know that Z_{p}[L]^{W_{V}} is a polynomial algebra. □

Theorem 5.3. Let p be an odd prime and let (W, L) be a finite Z_{p}-reflection group with the property that Z_{p}[L]^{W} is a polynomial algebra over Z_{p}. By Theorem 5.1 (W_{V} , L) is again a Z_{p}-reflection group and Z_{p}[L]^{W_{V}} is a polynomial algebra, for all non-trivial elementary abelian p-subgroups V ⊆ T̆ = L ⊗ Z/p^{∞}.

Assume that we know that for all such V there exists a p-compact group F (V) with discrete approximation to its maximal torus normalizer given by T̆⋊W_{V} such that F (V) is determined by N_{F (V)}, Aut(BF (V)) ≅ Aut(BN_{F (V)}), and H^{∗}(BF (V); Z_{p}) →^{≅} H^{∗}(B^{2}L; Z_{p})^{W_{V}}. Then there exists a p-compact group X with discrete approximation to its maximal torus normalizer given by T̆⋊W satisfying the same properties as listed for F (V).

Proof. Set N̆ = T̆⋊W. We want to construct a candidate “centralizer decomposition” diagram. Let A be the category with objects the non-trivial elementary abelian p-subgroup of T̆ and morphisms the homomorphisms between them induced by inclusion of subgroups and conjugation by elements in W. If V is a non-trivial elementary abelian p-subgroup of T̆, then by assumption there exists an, up to isomorphism unique, p-compact group F (V) with discrete approximation to the maximal torus normalizer j_{V} : CN̆ (V) → F (V) satisfying the assumptions listed in the theorem.

Let φ : V → V′ be a morphism in A, induced by conjugation by an element x ∈ W. We

now want to define a map F (V′) →^{F (φ)} F (V). Consider the diagram

[figure: V′; CN̆ (V′); c_{x^{−1}}; CN̆ (V); F (V′); F (V)]

Taking centralizer of the composite map x^{−1} : V′ → F (V) we get a space BC_{F (V)}(x^{−1}) = map(BV′, BF (V))_{Bx^{−1}}, which has discrete approximation to its maximal torus normalizer equal to CN̆ (V′). By assumption we get a unique (up to conjugacy) isomorphism F (V′) → C_{F (V)}(x^{−1}) under CN̆ (V′). By composing with the evaluation C_{F (V)}(x^{−1}) → F (V), we get a map F (φ) : F (V′) → F (V). We need to check that this gives us a well defined

functor from A^{op} to the homotopy category of spaces, i.e., that for V →^{φ} V′ →^{ψ} V′′, we have F (ψφ) ≅ F (φ)F (ψ). To see this consider the following diagram with obvious maps

\xymatrix{F (V) & C_{F (V)}(x^{−1}) \ar[l]_{ev} & F (V′) \ar[l]_{≅} & C_{F (V′)}(y^{−1}) \ar[l]_{ev} \ar@{-->}[dl]^{≅} & F (V′′) \ar[l]_{≅} \ar[ddll]^{≅} \\  &  & C_{C_{F (V )}(x^{−1})}({ \atop x^{−1}y^{−1}}) \ar@{-->}[ul]_{ev} \ar[d]^{≅} \\  &  & C_{F (V)}(x^{−1}y^{−1}) \ar[uull]^{ev}}

(Here ˜(·) denotes the adjoint map which is explained in Remark 6.4.) Note that the bottom composite from F (V′′) to F (V) is F (ψφ) and the top composite is F (φ)F (ψ). The top triangle is commutative, since the lower isomorphism in that triangle is just the map obtained by taking centralizers of the upper one. The rightmost square is homotopy commutative, since the corresponding square of isomorphisms between centralizers in N̆ is commutative, using our assumptions that maps are detected here. Finally, the leftmost square is homotopy commutative, by definition of the adjoint construction.

Hence we have constructed a well defined functor A → Ho(Spaces), where Ho(Spaces) is the homotopy category of spaces. By construction the functor obtained when taking cohomology of this diagram, can be identified with the canonical functor which on objects is given by V ↦ H^{∗}(BT̆; Z_{p})^{W_{V}}.

We want to lift this to a diagram in the category of topological spaces. The obstruction theory for doing this is described in [42, Thm. 1.1], noting that by [49, Lem. 11.15] our diagram is a so-called centric diagram so the assumptions of that theorem are satisfied.

By looking at their cohomology we see that all the spaces F (V) are connected and hence by [49, Thm. 7.5] have center given by T̆^{W_{V}}, since p is odd. In particular (see e.g. Lemma 10.2) the homotopy groups of ZF (V) are given by π_{0}(ZF (V)) = H^{1}(W_{V}; L) and π_{1}(ZF (V)) = L^{W_{V}}. By [47, §8] (for details see Section 10) lim_{V ∈A}^{∗} π_{∗}(F (−)) = 0, so by [42, Thm. 1.1] there exists a (unique) lift of our functor BF to a functor B̃F landing in Spaces. Set BX = (hocolim_{A} B̃F)_{p}^{∧}.

The spectral sequence for the calculating the cohomology of a homotopy colimit [17, Ch. XII 4.5] has E_{2}-term given as E_{2}^{i,j} = lim_{V ∈A}^{i} H^{j}(BT̆; Z_{p})^{W_{V}}. But again by [47, §8] these groups vanish for i > 0 and for i = 0 give lim_{A}^{0} H^{∗}(BT̆; Z_{p})^{W_{V}} ≅ H^{∗}(BT̆; Z_{p})^{W}. Hence the spectral sequence collapses onto the vertical axis, and we get H^{∗}(BX; Z_{p}) ≅ H^{∗}(BT̆; Z_{p})^{W}.

Since H^{∗}(BX; Z_{p}) is a polynomial algebra H^{∗}(X; Z_{p}) will be an exterior algebra on odd generators (cf. Theorem 12.1), so X is indeed a p-compact group. The fact that BX is determined by N and satisfies Aut(BX) ≅ Aut(BN), also follows easily from the above— the details are given in the proof of Lemma 6.3. □

Remark 5.4. Note that Theorem 5.3 in itself does not quite give a stand-alone proof of the realization and uniqueness of all center-free p-compact groups with Weyl group satisfying that Z_{p}[L]^{W} is a polynomial algebra, since (W_{V} , L) need not be center-free which prevents the obvious induction from working, the problem being the unitary groups.

6. Proof of the main theorem using Sections 8, 9, 10, 11, and 12

The purpose of this section is to prove the main Theorems 1.1 and 1.4, but in the proofs referring forward to Section 8 for information about elementary abelian p-subgroups of the Lie groups and to Section 10 for the obstruction group calculations.

We start by explaining the strategy in general terms. Recall that the centralizer of an elementary abelian p-subgroup ν : E → X of a p-compact group X is defined as the pcompact group C_{X}(ν) with classifying space BC_{X}(ν) = map(BE, BX)_{Bν}. It is a theorem of Dwyer-Wilkerson [48, Prop. 5.1 and 5.2] that this actually is a p-compact group and that the evaluation map to X is a monomorphism. A theorem of Dwyer-Zabrodsky [39] [74, Thm. 3.2] says that if G is a compact Lie group with component group a p-group, then the map

BC_{G}(ν(E))_{p}^{∧} → BC_{G_{p}^{∧}}(ν) = map(BE, BG_{p}^{∧})_{Bν}

induced by the adjoint of the canonical homomorphism E × C_{G}(ν(E)) → G is a homotopy equivalence. Note however that C_{X}(ν) is not in any natural way a subobject of X, and that the map to X is defined in terms of ν.

For a p-compact group X, let A(X) denote the Quillen category of X. The objects of A(X) are conjugacy classes of monomorphisms ν: E → X of non-trivial elementary abelian p-subgroups E into X. The morphisms (ν : E → X) → (ν′ : E′ → X) of A(X) consists of all group homomorphisms φ: E → E′ such that ν and ν′φ are conjugate.

The centralizer construction gives a functor

(6.1)  BC_{X} : A(X)^{op} → Spaces

that takes the monomorphism (ν : E → X) ∈ Ob(A(X)) to its centralizer BC_{X}(ν) = map(BE, BX)_{Bν} and a morphism φ to composition with Bφ : BE → BE′.

By a theorem of Dwyer-Wilkerson [49, §8], generalizing a theorem for compact Lie groups by Jackowski-McClure [73], the evaluation map

hocolim_{A(X)} BC_{X} → BX

induces an isomorphism on mod p homology. If X is connected center-free, then for all ν, the centralizer C_{X}(ν) is a p-compact group with smaller cohomological dimension setting the stage for a proof by induction. (The cohomological dimension of a p-compact group Y is defined as cd(Y) = max{n|H^{n}(Y; F_{p}) ≠ 0}; see [48, 6.13] and [50, 3.8].) To make use of this we need a way to construct a map from the elementary abelian p-subgroups and their centralizers in X, to another any other p-compact group X′ with the same maximal torus normalizer N.

Suppose that N is embedded in connected p-compact groups X and X′ via homomorphisms j and j′ respectively. If ν : E → X can be conjugated into T, i.e., if there exists μ : E → T such that μ composed with inclusion T → X equals ν, then μ is unique up to conjugation as a map to N by [50, Prop. 3.4], and furthermore by [49, Thm. 7.6], C_{N} (μ) is a maximal torus normalizer in C_{X}(ν). In this case j′μ will be an elementary abelian p-subgroup of X′, which we have assigned without making any choices, and C_{X′}(j′μ) will have maximal torus normalizer C_{N} (μ). Elementary abelian p-subgroups which can be conjugated into T are called toral subgroups, and elementary abelian p-subgroups which do not have this property are called non-toral subgroups. The problem hence arises how to compare the centralizers in the case of non-toral elementary abelian p-subgroups. This problem was addressed by the third-named author in [90]:

Theorem 6.1. [90] Let X be a p-compact group with maximal torus normalizer N.

If ν : E → X is an elementary abelian p-subgroup of X, then there exists a lift μ : E → N with ν = jμ such that C_{N} (μ) → C_{X}(ν) is a maximal torus normalizer. Furthermore, if E′ ⊆ E and μ′ is a lift of ν_{|E′} with the above property, then μ′ can be extended to a lift μ of E which also has that property.

Assume X is connected. If ν : E → X has rank one then ν is toral [48, Prop. 5.5.] and the above lift μ is unique up to conjugation in N. If ν has rank two, then the lift is unique if ν is toral and if ν is non-toral then there are precisely p + 1 different lifts with the further property that π_{0}(μ) : E → π_{0}(N) is not injective, corresponding to the p + 1 rank one subgroups of E. □

Remark 6.2. The analogous theorem of the above in the classical case of compact Lie groups does not seem to appear in the literature. However, as was suggested to the thirdnamed author by H. R. Miller and subsequently explained in detail by J.-P. Serre [112], this can be obtained by a modification of the proof of [119, Thm. 5.16]—strictly speaking, we shall only need this theorem in the cases where either X is the p-completion of a compact Lie group, or where the mod p cohomology of BX is a polynomial algebra, the latter case being trivial.

If X is assumed connected then we can always arrange that the lift in Theorem 6.1 furthermore satisfies that the kernel of the map π_{0}(μ) : E → π_{0}(N) is non-trivial, and such a map will be called a preferred lift. Note that if ν : E → X can be factored through T then the corresponding map μ : E → N is a preferred lift and it is furthermore unique up to conjugacy in N, i.e., the preferred lift is exactly the factorization through the maximal torus described earlier (see also [90, Prop. 4.10]).

We now want to show how to use Theorem 6.1 to construct a map between the centralizer diagrams of connected p-compact groups X and X′.

Suppose that ν : V → X has rank one and that C_{X}(ν) is determined by N_{C_{X}(ν)} and satisfies Aut(BC_{X}(ν)) ≅ Aut(BN_{C_{X}(ν)}). Let μ be a preferred lift of ν. Since ν is rank one and X is connected the unique preferred lift μ will in fact factor through T by [48, Prop. 5.5], and μ is a preferred lift of j′μ by [49, Thm. 7.6]. Hence C_{N} (μ) is a maximal torus normalizer for both C_{X}(ν) and C_{X′}(j′μ), so by assumption there exists a homomorphism h_{ν}, unique up to conjugacy, making the diagram

(6.2)  \xymatrix{ & C_{N} (μ) \ar[dr]^{j′} \ar[dl]_{j} \\ C_{X}(ν) \ar[rr]^{h_{ν}}_{≅} &  & C_{X′}(j′μ)}

commute. For an arbitrary elementary abelian p-subgroup ν : E → X we can restrict to a rank one subgroup V, and all the corresponding maps in the general case by viewing things as going on inside the centralizer of the rank one subgroup. However it is not a priori clear that this construction does not depend on the choice of the rank one subgroup V. Furthermore one has the problem that this procedure only gives a map of diagrams in the homotopy category, and one then needs to prove that this diagram can be rigidified to a diagram in the category of spaces. The next lemma and its proof makes this construction precise and states what needs to be checked—the calculations to verify that these conditions are indeed verified for all p-compact groups is essentially the contents of the rest of the paper.

Lemma 6.3. Let X and X′ be two connected p-compact groups with the same maximal torus normalizer N embedded via j and j′ respectively. Assume that for all rank one elementary abelian p-subgroups ν : E → X of X the centralizer C_{X}(ν) is determine by N_{C_{X}(ν)} and that Aut(BC_{X}(ν)) ≅ Aut(BN_{C_{X}(ν)}) when ν is of rank one or two.

(1) Assume that for every rank two non-toral object ν : E → X with a preferred lift μ, neither the conjugacy class of j′μ : E → X′ nor the conjugacy class of the induced map h_{ν,V} : C_{X}(ν) → C_{X′}(j′μ) induced by taking “adjoints relative to the rank one subgroup ν_{|V} : V → X” (see diagram (6.3)) depends on the p + 1 choices of a rank one subgroup V in E. Then there exists a map in the homotopy category of spaces from the centralizer diagram of BX to BX′ (seen as a constant diagram), i.e., an element in lim_{ν∈A(X)}^{0}[BC_{X}(ν), BX′], given via maps h_{ν,V} as hinted above.

(2) Assume furthermore lim_{ν∈A(X)}^{i}π_{j}(BZC_{X}(ν)) = 0 for j = 1, 2 and i = j, j + 1, then there is a unique lift of this to a map in the (diagram) category of spaces, i.e., there exists a unique equivalence f: X → X′ under N and Aut(BX) ≅ Aut(BN).

Before the proof, let us recall the construction of adjoint maps, since these play a central role in the proof:

Remark 6.4 (Adjoint maps). Let A be an abelian p-compact group, X be a p-compact group, and ν : A → X be a homomorphism. Suppose that E is a subgroup of A and note that we have a canonical map

Bψ : BA × BE →^{mult} BA → BX

which only depends on the conjugacy class of ν. Since furthermore

π_{0}(map(BA × BE, BX)) = ∐_{ξ∈[BE,BX]} π_{0}(map(BA, map(BE, BX)_{ξ}))

we get that every element ν : A → X gives rise to an element ν̃ : A → C_{X}(ν_{|E}) making the diagram

\xymatrix{ & C_{X}(ν_{|E}) \ar[d]^{ev} \\ A \ar[ur]^{ν̃} \ar[r]^{ν} & X}

commutative. Here ν̃ is well defined up to conjugacy, from the conjugacy class of ν. We will always use the notation ˜(·) for this construction.

Proof of Lemma 6.3. Let μ be a preferred lift of ν. Since X is connected we can by Theorem 6.1 choose a rank one subgroup V in the kernel of E → π_{0}(N). Since μ_{|V} factors through T it follows from [49, Thm. 7.6] that μ_{|V} is a preferred lift of both ν_{|V} and j′μ_{|V}. By assumption C_{X}(ν_{|V}) is determined by N_{C_{X}(ν_{|V} )} and Aut(BC_{X}(ν_{|V})) ≅ Aut(BN_{C_{X}(ν_{|V} )}) so we get a commutative diagram

\xymatrix{ & C_{N} (μ_{|V}) \ar[dl] \ar[dr] \\ C_{X}(ν_{|V}) \ar[rr]^{h_{ν_{|V}}}_{≅} &  & C_{X′}(j′μ_{|V})}

The adjoint maps of ν, μ, and j′μ (see Remark 6.4) map into the diagram in a coherent way, which expresses j′μ in terms of ν and the rank one subgroup V, so it only depends on those parameters. Taking further adjoints with respect to ν, μ and jμ produces a commutative diagram

(6.3)  \xymatrix{ &  & C_{N} (μ) \ar[ddll]_{j} \ar[ddrr]^{j′} \\  &  & C_{C_{N} (μ_{|V} )}(μ̃) \ar[u]_{≅} \ar[dl] \ar@{-->}[dr] \\ C_{X}(ν) & C_{C_{X}(ν_{|V} )}(ν̃) \ar[l]^{≅} \ar[rr]_{≅}^{h̃_{ν_{|V}}} &  & C_{C_{X′}(j′μ_{|V} )}(j̃′μ) \ar[r]_{≅} & C_{X′}(j′μ)}

where h̃_{ν_{|V}} is the map induced from h_{ν_{|V}} on the centralizers. But this means that C_{N} (μ) is a maximal torus normalizer also in C_{X}(j′μ), e.g., by the characterizing property for maximal torus normalizers give in [90, Thm. 1.2(3)], so in particular μ is a preferred lift of j′μ.

Call the bottom left-to-right composite in the above diagram for h_{ν,V}. We want to see that this map does not depend on V (and we will then drop the V from the notation).

Note that by the above diagram (6.3) the composition E →^{ν̃} C_{X}(ν) →^{h_{ν,V}} C_{X′}(jμ) → X′ equals j′μ. By a slight abuse of notation we denote this map h_{V} (ν).

If ν has rank two and is toral then h_{ν,V} does not depend on the choice of V by diagram (6.3) since μ is uniquely determined from ν in this case, and we are assuming that centralizers of rank two subgroups have the same automorphisms as their maximal torus normalizer. If ν has rank two and is non-toral, then we are simply assuming that h_{ν,V} does not depend on V.

We want to see that the rank two condition forces this to hold in general. Let V_{1} and V_{2} be two different rank one subgroups of E, and set U = V_{1} ⊕ V_{2}. Since h_{ν_{|U} ,V} does not depend on the choice of V the following diagram commutes up to conjugation

\xymatrix{ & C_{X}(ν_{|V_{1}}) \ar[r]^{h_{ν_{|V_{1}}}} & C_{X′}(h(ν_{|V_{1}})) \ar[dr] \\ E \ar[ur]^{ν̃} \ar[r]^{ν̃} \ar[dr]^{ν̃} & C_{X}(ν_{|U}) \ar[r]^{h_{ν_{|U}}} \ar[u] \ar[d] & C_{X′}(h(ν_{|U})) \ar[u] \ar[d] \ar[r] & X′ \\  & C_{X}(ν_{|V_{2}}) \ar[r]^{h_{ν_{|V_{2}}}} & C_{X′}(h(ν_{|V_{2}})) \ar@{-->}[ur]}

But this shows that in general h_{ν,V} does not depend on the choice of V, since for any two choices of rank one subgroups V_{1} and V_{2} the above diagram shows that both maps will be centralizers of the common h_{ν_{|U}}.

It is now easy to construct a map from the centralizer diagram of X into X′, by to ν assigning the homomorphism C_{X}(ν) →^{h_{ν}} C_{X′}(h(ν)) → X′. We have to check that this is really a natural transformation, i.e., that we get commutative diagrams on morphisms. Since all morphisms in A(X) are the composite of an isomorphism followed by an inclusion, it is enough to check on these. Suppose φ : (ν : E → X) → (ν′ : E′ → X) is an isomorphism. If ν has rank one then by assumption C_{X}(ν) has the same automorphisms as its maximal torus normalizer so the diagram

\xymatrix{C_{X}(ν′) \ar[r]^{≅}_{h_{ν′}} \ar[d]^{≅}_{C_{X}(φ)} & C_{X′}(j′μ′) \ar[d]_{≅}^{C_{X′}(φ)} \\ C_{X}(ν′φ) \ar[r]^{h_{ν′φ}}_{≅} & C_{X′}(j′μφ)}

commutes up to conjugation, since we can view the diagram of isomorphisms as taking place under C_{N} (μ′) → C_{N} (μ′φ), for a preferred lift μ′ of ν′. For a general isomorphism, if restrict to a rank one subgroup V in E and its image φ(V) in E′, then the diagram above is, by construction of h_{ν′} and h_{ν′φ}, just obtained by taking centralizers of the adjoint maps of ν′φ and ν′ of the diagram in rank one case, and it hence has to commute up to conjugation as well. If φ is an inclusion of a subgroup then above diagram will also commute up to conjugation, by the same adjoint argument. This shows that for all maps φ in A(X) the diagram

\xymatrix{C_{X}(ν′) \ar[dr] \ar[rr]^{C_{X}(φ)} &  & C_{X}(ν) \ar[dl] \\  & X′}

commutes up to conjugation, where the diagonal maps are the natural composites C_{X}(ν) →^{h_{ν}} C_{X′}(h_{ν}(ν)) → X′. So we have constructed a map, of diagrams in the homotopy category, from the centralizer diagram of X to X′ (seen as a constant diagram), or in other words we have given an element

[ρ] ∈ lim_{ν∈A(X)}^{0}π_{0}(map(BC_{X}(ν), BX′))

Letting e denote the evaluation C_{X′}(h_{ν}(ν)) → X′ we have by [51, Rem. after Def. 6.3] [49, 11.15] (which says that the centralizer diagram for a p-compact group is “centric”) that

\xymatrix{map(BC_{X′}(h_{ν}(ν)), BC_{X′}(h_{ν}(ν)))_{1} \ar[r]^{e_{#}}_{≃} & map(BC_{X′}(h_{ν}(ν)), X′)_{e} \ar[r]^{^{h_{ν}^{#}}}_{≃} & map(BC_{X}(ν), X′)_{eh_{ν}}}

Likewise, by the homotopy equivalences established earlier in the proof

map(BC_{X′}(h_{ν}(ν)), BC_{X′}(h_{ν}(ν)))_{1} ≃ map(BC_{X}(ν), BC_{X}(ν))_{1}

which by the definition of the center [49] equals BZC_{X}(ν). Since these identifications are natural, this gives a canonical identification of the functor ν ↦ π_{i}(map(BC_{X}(ν), BX′)_{[ρ]}) with ν ↦ π_{i}(BZC_{X}(ν)).

By obstruction theory (see [131, Prop. 3] [76, Prop. 1.4]) the existence obstructions for lifting this to an element in π_{0}(holim_{A(X)} map(BC_{X}(ν), X′)) ≅ π_{0}(map(BX, BX′)) lie in

lim_{ν∈A(X)}^{i+1}π_{i}(map(BC_{X}(ν), BX′)_{[ρ]}) ≅ lim_{ν∈A(X)}^{i+1}π_{i}(BZC_{X}(ν)), i ≥ 1

But by assumption all these groups are identically zero, our data lifts to a map Bf : BX → BX′.

We now want to see that the construction of f forces it to be an isomorphism. Let N_{p} denote a p-normalizer of T, i.e., the union of components in N corresponding to a Sylow p-subgroup of W. Since N_{p} has non-trivial center (since the action of a p-group on T̆ has to have a fixed point), we can find a rank one central subgroup μ : V → N_{p}, and so we can view N_{p} as sitting inside C_{N} (μ). Hence by construction the diagram

\xymatrix{ & N_{p} \ar[dl]_{j} \ar[dr]^{j′} \\ X \ar[rr]^{f} &  & X′}

commutes up to conjugation, and in particular fj : N_{p} → X′ is a monomorphism. This easily implies that f is a monomorphism as well: We have that H^{∗}(BN_{p}) if finitely generated over H^{∗}(BX′) via H^{∗}(Bfj) by [48, Prop. 9.11]. By an application of the transfer [48, Thm. 9.13] the map H^{∗}(Bj) : H^{∗}(BX) → H^{∗}(BN_{p}) is a monomorphism, and since H^{∗}(BX′) is noetherian by [48, Thm 2.3] we conclude that H^{∗}(BX) is finitely generated over H^{∗}(BX′) as well, by the definition of noetherian. Hence f : X → X′ is a monomorphism by another application of [48, Prop. 9.11]. Since we can identify the maximal torus of X′ and X, the definition of the Weyl group defines a map between the Weyl groups W_{X} → W_{X′}, which has to be injective since the Weyl groups act faithfully on T (by [48, Thm. 9.7]). But since we know that X and X′ have the same maximal torus normalizer, the above map of Weyl groups has to be an isomorphism. By [49, Thm. 4.7] (or [86, Prop. 3.7] using [48, Thm. 9.7]) this means that f is indeed an isomorphism.

We now want to argue that this f is a map under N. By Lemma 2.1 we know that there exists a unique g ∈ π_{0}(Aut(BN)) such that

\xymatrix{N \ar[r]^{g} \ar[d]_{j} & N \ar[d]^{j′} \\ X \ar[r]^{f} & X′}

commutes up to conjugation. By covering space theory and Sylow’s theorem we can restrict g to a self-map of g′ making the diagram

\xymatrix{N_{p} \ar[r]^{g′} \ar[d]_{j} & N_{p} \ar[d]^{j′} \\ X \ar[r]^{f} & X′}

commute. Furthermore any other map N_{p} → N_{p} fitting in this diagram will be conjugate to g′ in N, by the proof of Lemma 2.1. However by construction f is a map under N_{p} so g′ is conjugate in N to the canonical inclusion. But by Proposition 3.1 and 3.2 we get that if an outer automorphism of N restricts to the identity on N_{p} then it is in fact the identity. This shows that g = 1, i.e., that f is a map under N. This also shows that π_{0}(Aut(BX)) → π_{0}(Aut(BN)) is surjective, since for any automorphism g : N → N, jg is also a maximal torus normalizer in X by [90, Thm. 1.2(3)].

Note that if Aut_{1}(BN) is not contractible then we can find rank one p-subgroup ν : V → T such that C_{N} (ν) →^{≅} N which by assumption means that Aut(BX) ≅ Aut(BN). So we can assume that Aut_{1}(BN) is contractible in which case Aut_{1}(BX) is as well by [49, Thm. 1.3 and Thm. 7.5].

The only remaining claim in the lemma is that the map π_{0}(Aut(BX)) → π_{0}(Aut(BN)), defined in Lemma 2.1, is injective under the additional assumption that lim_{ν∈A(X)}^{i} π_{i}(BZC_{X}(ν)), i ≥ 1. In other words we have to see that any self-equivalence f of X which, up to conjugacy, induces the identity on N is in fact conjugate to the identity. But if we examine the above argument with X′ = X, the map on centralizers of rank one elements induced by f has to be the identity by the rank one uniqueness assumption. The maps for higher rank are centralizers of maps of rank one, so they as well have to be the identity. Hence f maps to the same element as the identity in lim_{ν∈A(X)}^{0} π_{0}(map(BC_{X}(ν), BX)), which means that f actually is the identity by the vanishing of the obstruction groups (again, see [131, Prop. 4] or [76, Prop. 1.4]). □

Remark 6.5. Note how the assumptions of the lemma fail (as they should) for group SO(3) at the prime 2 which is not determined by its normalizer. In this case the element diag(−1, −1, 1) in the maximal torus SO(2) × 1 will be fixed under the Weyl group action and have centralizer equal to the maximal torus normalizer O(2).

Remark 6.6. It is perhaps natural to ask whether the preferred lift of rank two elementary abelian p-subgroups is unique in the “subgroup of Lie group sense”, i.e., whether the Weyl group of ν : E → X acts transitively on the rank one subgroups of E. The lists in Section 8 and 9 reveals that this is false for the subgroup E_{E_{6}}^{2a} of the group E_{6} but true for all other center-free simple compact Lie groups at odd primes.

Lemma 6.7. Let X and X′ be two connected p-compact groups with the same maximal torus normalizer N embedded via j and j′ respectively. Assume that for all rank one elementary abelian p-subgroups η : E → X of X the centralizer C_{X}(η) is determined by N_{C_{X}(η)} and that Aut(BC_{X}(η)) ≅ Aut(BN_{C_{X}(η)}) when η is of rank one and two.

Let ν : E → X be a rank two elementary abelian p-subgroup of X and assume that:

(1) The Weyl group W (ν) of ν contains SL(E).

(2) ν is up to isomorphism the only (non-toral) elementary abelian p-subgroup whose centralizer is isomorphic to C_{X}(ν).

Then the map j′μ : E → X′ constructed by picking a preferred lift μ : E → N of ν does not depend on the p + 1 choices of μ (corresponding to the p + 1 rank one subgroups of E; see Theorem 6.1 and Lemma 6.3).

Proof. Let {ν} denote the orbit of ν in [BE, BX] under the natural GL(E)-action and note that the order of the set {ν} equals the index of W (ν) in GL(E). Let {μ} denote the set of the preferred lifts in [BE, BN] which lifts elements in {ν}. Since any ν has p + 1 preferred lifts the map of GL(E)-sets {μ} → {ν} induced by j is surjective and (p + 1)-to-1.

By assumption SL(E) ⊆ W (ν) so the Weyl group W (ν) acts transitively on the rank one subgroups of E. Hence if μ is a preferred lift of ν, any other preferred lift of μ can be obtained as μα for some α ∈ W (ν). In other words also {μ} is a transitive GL(E)-set.

We now want to argue that {μ} is also exactly the preferred lifts of the GL(E)-orbit {j′μ} of j′μ where μ is any lift of ν. Namely, if μ′ is any preferred lift of j′μ then by the first assumptions in the Lemma (about centralizers of rank one and two subgroups) μ′ is also a preferred lift of jμ′ and C_{X}(jμ′) is isomorphic to C_{X′}(j′μ) and C_{X}(ν) (see the proof of Lemma 6.3). Hence by the assumption that there is only one object up to conjugacy in X with centralizer isomorphic to C_{X}(ν) there exists an α ∈ GL(E) such that ν = jμ′α. In other words μ′ ∈ {μ}. So surjective map of GL(E) sets {μ} → {j′μ} is (p + 1)-to-1 as well.

Hence the set {j′μ} has the same cardinality as {ν}. In particular the stabilizer of j′μ is a subgroup of GL(E) of index [GL(E) : W (ν)]. But since SL(E) ⊆ W (ν) this means that this stabilizer equals W (ν), since W (ν) is the unique subgroup with this index.

In particular j′μ = (j′μ)α = j′(μα) and since μα exhausts the preferred lifts of ν, we conclude that j′μ does not depend on the choice of preferred lift. □

Lemma 6.8. Let X and X′ be two connected p-compact groups with the same maximal torus normalizer N embedded via j and j′ respectively. Assume that for all rank one and two elementary abelian p-subgroups η : E → X of X the centralizer C_{X}(η) is determined by N_{C_{X}(η)} and that Aut(BC_{X}(η)) ≅ Aut(BN_{C_{X}(η)}). Furthermore assume that for η : E → X of rank two, self-equivalences of C_{X}(η)_{1} are detected by its maximal torus (which is covered by the previous assumption if p is odd; cf. Proposition 3.2).

Let ν : E → X be a rank two non-toral elementary abelian p-subgroup of X. Assume the following:

(1) The Weyl group W (ν) of ν acts transitively on the rank one subgroups of E.

(2) E × C_{X}(ν)_{1} →^{≅} C_{X}(ν), where the identity component C_{X}(ν)_{1} is a non-trivial pcompact group and α ∈ W (ν) acts on E × C_{X}(ν)_{1} as α × 1.

Then neither jμ nor the map h_{ν,V} : C_{X}(ν) → C_{X′}(j′μ) of Lemma 6.3 depends on the choice of the rank one subgroup V of E (i.e., the assumptions of Lemma 6.3(1) are satisfied).

Proof. Fix a preferred lift μ of ν, and set T_{E} equal to the connected component of the space C_{N} (μ) (which will be a maximal torus in C_{X}(ν) and by assumption non-trivial). Let V be the kernel of π_{0}(μ) : E → π_{0}(N).

Since the map μ : E → C_{N} (μ) is central, so we get an induced map μ̄ : E × T_{E} = E × C_{N} (μ)_{1} → C_{N} (μ) → N.

The assumption about the action means that for α ∈ W (ν) we have the following up to conjugation commutative diagram:

\xymatrix{E × T_{E} \ar[r] \ar[d]^{α×1} & C_{N} (μ) \ar[r] & C_{X}(jμ) \ar[d]^{α} \ar[dr] \\ E × T_{E} \ar[r] & C_{N} (μ) \ar[r] & C_{X}(jμ) \ar[r] & X}

so jμ̄(α × 1) is homotopic to jμ̄.

Let U be a rank one elementary abelian p-subgroup of T_{E}. By assumption we have a unique map hμ̄_{|U} making the following diagram commutative up to conjugation.

\xymatrix{ & E × T_{E} \ar[ddl]_{j̃μ̄} \ar[d]^{˜μ̄} \ar[ddr]^{j̃′μ̄} \\  & C_{N} (μ̄_{|U}) \ar@{-->}[dr] \ar[dl] \\ C_{X}(jμ̄_{|U}) \ar[rr]^{hμ̄_{|U}}_{≅} \ar[d] &  & C_{X′}(j′μ̄_{|U}) \ar[d] \\ X &  & X′}

so j′μ(α × 1) is also homotopic to j′μ. The same is hence true for the adjoint maps E × T_{E} → C_{X}(ν) and E × T_{E} → C_{X′}(j′μ).

\xymatrix{ & C_{N} (μ) \ar[dl]_{j} \ar[dr]^{j′} & E × T_{E} \ar[l]_{˜μ̄} \ar[r]^{{ \atop μ̄(α×1)}} & C_{N} (μα) \ar[dl]_{j′} \ar[dr]^{j} \\ C_{X}(ν) \ar[rr]_{h_{ν,V}}^{≅} &  & C_{X′}(j′μ) &  & C_{X}(ν) \ar[ll]^{h_{ν,α^{−1}(V )}}_{≅}}

which is up to conjugation commutative by the above and where the two maps E × T_{E} → C_{X}(ν) are identical.

Hence h_{ν,α^{−1}(V)}^{−1}h_{ν,V} is a self-equivalence of C_{X}(ν) ≅ E × C_{X}(ν)_{1} which is the identity on E × T_{E}. Since by assumption self-maps of C_{X}(ν)_{1} are determined by their restriction to a maximal torus we hence conclude that h_{ν,V}^{−1} h_{ν,α^{−1}(V)} is indeed homotopic to the identity (cf. [49, Lem. 5.3]) and thus h_{ν,V} does not depend on the choice of subgroup V in E. □

The above two lemmas are in fact general enough to let us verify the conditions of Lemma 6.3 in all the situations we shall encounter, except the subgroup E_{E_{6}}^{2a} of E_{6} where we need a small variation of Lemma 6.8.

Lemma 6.9. Let ν : E_{E_{6}}^{2a} ↪ X = E_{6}_{3}^{∧} be the rank two non-toral elementary abelian 3subgroup of E_{6} listed in Theorem 8.9. Assume that for all rank one elementary abelian p-subgroups η : E → X of X the centralizer C_{X}(η) is determine by N_{C_{X}(η)} and that Aut(BC_{X}(η)) ≅ Aut(BN_{C_{X}(η)}) when η is of rank one or two.

Then the conjugacy class of j′μ : E → X′ as well as the conjugacy class of h_{ν,V} : C_{X}(ν) → C_{X′}(j′μ) do not depend on the choice of rank one subgroup of V of E.

Proof. The proof is almost identical to Lemma 6.8 but where we have to argue a bit more carefully since W (ν) does not act transitively on the rank one subgroups of E.

Let μ_{1}, …, μ_{4} be the four preferred lifts. By Proposition 8.10 we have that T_{E} = C_{N} (μ_{i}(E))_{1} is independent of the choice of μ. (Note that we are taking centralizer in the Lie group sense.) Set μ̄_{i} : E × T_{E} →^{μ_{i}×1} N. Also by Proposition 8.10 the conjugacy class of jμ̄_{i} does not depend on μ_{i}. This brings us in exactly the same situation as in the proof of Lemma 6.8 (with the μ̄(α × 1) replaced by the μ̄_{i}), which proves the lemma. □

Proof of Theorem 1.4, and 1.1 using Section 8, 9,10, 11, and 12. We will show that Theorem 1.1 and 1.4 both holds by a simultaneous induction on the cohomological dimension of X. We will furthermore add to the statement we are doing induction over the statement that if X is connected and Z_{p}[L]^{W} is a polynomial ring, then H^{∗}(BX; Z_{p}) ≅ H^{∗}(BT; Z_{p})^{W}.

By Lemma 4.3 and 4.5, Theorem 1.4 holds for a p-compact group X if it holds for its connected component X_{1}, so we can assume that X is connected.

By [5, Thm. 1.2], if (W, L) is realized of as the Weyl group of a p-compact group, then N_{X} will be split i.e. N̆_{X} ≅ T̆⋊W_{X} (cf. also [124], [54], and [93] for the Lie case). So, to prove Theorem 1.4 and 1.1 we have to show that given any finite Z_{p}-reflection group (W, L) there exists a unique connected p-compact group X realizing (W, L), with self-maps satisfying Aut(BX) ≅ Aut(BN_{X}) (since this implies π_{0}(Aut(BX)) ≅ N_{GL(L_{X})}(W)/W by Proposition 3.1 and 3.2).

If (W, L) can be written as a non-trivial product, then there exists a p-compact group X realizing (W, L) by the induction assumption. The p-compact group is unique by [50, Thm. 1.4] and the induction assumption, i.e., X is determined by N_{X}. Lemma 4.1 and the induction assumption shows that X has the right space of automorphisms. Hence we can assume that (W, L) cannot be written as a product.

By the classification of finite Z_{p}-reflection groups (Theorem 11.1), (W, L) is either of the form (W_{G}, L_{G} ⊗Z_{p}), for some connected compact Lie group G, or (W, L) is one of the exotic Z_{p}-reflection groups.

In the first case (W, L) can of course be realized by X = G_{p}^{∧}. Note that, by Theorem 12.2 (and Theorem 12.1) H^{∗}(BX; Z_{p}) ≅ H^{∗}(BT; Z_{p})^{W} if and only if Z_{p}[L]^{W} is a polynomial algebra.

In the latter case Z_{p}[L]^{W} is a polynomial algebra by Theorem 12.2 and (W, L) satisfies T̆^{W} = 0 by Theorem 11.1. Hence induction shows that the assumptions of Theorem 5.3 are satisfied. So, by Theorem 5.3 there exists a unique p-compact group X with Weyl group (W, L) which is determined by N_{X}, and satisfies that Aut(BX) ≅ Aut(BN) and H^{∗}(BX; Z_{p}) ≅ H^{∗}(BT; Z_{p})^{W}.

We now want to show that X is uniquely determined by (W_{X}, L_{X}) and satisfies the properties about self-maps listed in Theorem 1.1 and 1.4. In fact, the statement about self-maps in Theorem 1.1 follows from the one in Theorem 1.4 by Proposition 3.2, so we can concentrate on Theorem 1.4.

By Lemma 4.7 we can assume that X is center-free. Likewise by the splitting theorem [50, Thm. 1.4 and 1.5] together with Lemma 4.1 we can assume that X is simple. By Theorem 12.2 (W, L) either has the property that Z_{p}[L]^{W} is a polynomial algebra, or (W, L) is one of the reflection groups (W_{P U(n)}, L_{P U(n)} ⊗ Z_{p}) (with p | n), (W_{E_{8}}, L_{E_{8}} ⊗ Z_{5}), (W_{F_{4}}, L_{F_{4}} ⊗ Z_{3}), (W_{E_{6}}, L_{E_{6}} ⊗ Z_{3}), (W_{E_{7}}, L_{E_{7}} ⊗ Z_{3}), or (W_{E_{8}}, L_{E_{8}} ⊗ Z_{3}).

If the Z_{p}-reflection group (W, L) has the property that Z_{p}[L]^{W} is a polynomial algebra, then we have just seen that there exists a space X′ such that H^{∗}(BX′; Z_{p}) ≅ H^{∗}(B^{2}L; Z_{p})^{W}.

Hence all elementary abelian p-subgroups of X′ are toral (cf. Lemma 7.8). In particular X′ has no rank two non-toral elementary abelian p-subgroups so the assumptions of Lemma 6.3(1) (used with X and X′ switched around) are satisfied. Likewise by Theorem 10.1 the assumptions of Lemma 6.3(2). Hence Lemma 6.3 implies that there exists an equivalence of p-compact groups X′ → X, and that X satisfies the conclusion of Theorem 1.4.

Consider (W, L) = (W_{P U(n)}, L_{P U(n)} ⊗ Z_{p}) where p | n. We divide up in two cases. If n = pk with k > 1 then Theorem 9.1 shows that the assumptions of Lemma 6.8 are satisfied which implies that the assumptions of Lemma 6.3(1) are satisfied. By Theorem 10.1 the assumptions of Lemma 6.3(2) are also satisfied, so Lemma 6.3 produces an equivalence of p-compact groups P U(n)_{p}^{∧} → X, and shows that X satisfies the conclusion of Theorem 1.4. If n = p then we argue as above but replace the reference to Lemma 6.8 with a reference to Lemma 6.7, noting that the statement about h_{ν,V} in this case follows for free from that of j′μ, since E ≅ C_{P U(p)}(E).

If (W, L) = (W_{E_{8}}, L_{E_{8}} ⊗ Z_{5}) then by Theorem 8.2(3) E_{8} does not have any non-toral rank 2 elementary abelian 5-subgroups. Furthermore the higher limits obstructions which feature in the assumptions for Lemma 6.3(2) vanish by Theorem 10.1. Hence Lemma 6.3 implies that there exists an equivalence of p-compact groups E_{8}_{5}^{∧} → X, and that X satisfies the conclusion of Theorem 1.4.

Likewise if (W, L) equals (W_{F_{4}}, L_{F_{4}} ⊗ Z_{3}), (W_{E_{7}}, L_{E_{7}} ⊗ Z_{3}), or (W_{E_{8}}, L_{E_{8}} ⊗ Z_{3}), then Theorem 8.2(3) shows that there are no rank two non-toral elementary abelian 3-subgroups and the obstruction groups vanish by Theorem 10.1, so Lemma 6.3 again implies that X is homotopy equivalent to the p-completion of the appropriate Lie group and satisfies the conclusion of Theorem 1.4.

Finally, if (W, L) = (W_{E_{6}}, L_{E_{6}} ⊗ Z_{3}) then the rank 2 non-toral subgroup E_{E_{6}}^{2b} of Theorem 8.9 satisfies the conditions of Lemma 6.8 so the assumptions of Lemma 6.3(1) are satisfied for this subgroup. Likewise for the subgroup E_{E_{6}}^{2a} the custom made Lemma 6.9 shows that the assumptions of Lemma 6.3(1) are also satisfied for this subgroup. Since the assumptions of Lemma 6.3(2) are satisfied by Theorem 10.1 we conclude by Lemma 6.3 that X is homotopy equivalent to E_{6}_{3}^{∧} and satisfies the conclusion of Theorem 1.4 also in this case. This concludes the proof of the main theorem. □

Remark 6.10. Note that taking the case (W_{E_{6}}, L_{E_{6}} ⊗ Z_{3}) last in the above theorem is a bit misleading, since groups with adjoint form E_{6} appear as centralizers in E_{7} and E_{8}, so a separate inductive proof of uniqueness in those case would require knowledge of uniqueness of E_{6}.

Remark 6.11. The very careful reader might have noticed that the splitting result in [5], which we use in the above proof, to conclude the splitting for (W_{P U(3)}, L_{P U(3)}) refers to a uniqueness result in [20]. We now quickly sketch a more direct way to see this, which we were told by Dwyer-Wilkerson: We need to see that a 3-compact group with Weyl group (W_{P U(3)}, L_{P U(3)} ⊗ Z_{3}) has to have split maximal torus normalizer N. So, suppose that X is a hypothetical 3-compact group like above but with non-split maximal torus normalizer. By a transfer argument (cf. [48, 9.12]) N_{3} has to be non-split as well. Since every elementary abelian 3-subgroup in X can be conjugated into N_{3} (since χ(X/N_{p}) is prime to p), this means that all elementary abelian 3-subgroups in X are toral. Furthermore by [50, Prop. 3.4] conjugation between toral elementary abelian p-subgroups is controlled by the Weyl group, so the Quillen category of X in fact agrees with the Quillen category of N. The category has up to isomorphism one element of rank two and two of rank one. The centralizers C_{N} (V) of these are respectively T, T : Z/2, and T · Z/3. The unique 3-compact groups corresponding to these centralizers are in fact given by BC_{N} (V)_{3}^{∧}. Hence the map BN → BX is an equivalence by the centralizer cohomology decomposition theorem [49, §8]. But since N is non-split, we can find a map Z/p^{2} → N, which is not conjugate in N to a map into T. Hence the corresponding map Z/p^{2} → X cannot be conjugated into T either, contradicting [48, Prop. 5.5].

7. Consequences of the main theorem

In this section we prove the theorems listed in the introduction which are consequences of the main theorem. We also give a couple of remarks relating our method to other possible approaches.

Proof of Theorem 1.2. The theorem follows directly from Theorem 1.1 together with the classification of finite Z_{p}-reflection groups (Theorem 11.1). □

Proof of Theorem 1.5. By [86, Thm. 1.4] X is equivalent to a p-compact group of the form (X′ × T′′)/A, where X′ is a simply connected p-compact group, T′′ is a p-compact torus, and A is a finite central subgroup of the product. Hence we have X/T ≅ X′/T′, where T and T′ are maximal tori of X and X′ respectively. So we can without restriction assume that X is simply connected.

For compact connected Lie groups the statement of this theorem is the celebrated result of Bott [12, Thm. A]. Hence by Theorem 1.2 it is enough to prove the theorem when X is an exotic p-compact group. In that case H^{∗}(BX; Z_{p}) is a polynomial algebra with generators in even degrees, and number of generators equal the rank of X (by the proof of Theorem 1.4). The same is true over F_{p}, and since H^{∗}(BT; F_{p}) is finitely generated over H^{∗}(BX; F_{p}) by [48, Prop. 9.11], we have that H^{∗}(BX; F_{p}) → H^{∗}(BT; F_{p}) has to be injective by a Krull dimension consideration. But since they are both polynomial algebras it follows by e.g., [52, §11] that H^{∗}(BT; F_{p}) is in fact free over H^{∗}(BX; F_{p}). Hence the Eilenberg-Moore spectral sequence of the fibration X/T → BT → BX collapses and

H^{∗}(X/T; F_{p}) ≅ F_{p} ⊗_{H^{∗}(BX;F_{p})} H^{∗}(BT; F_{p}).

In particular H^{∗}(X/T; F_{p}) is concentrated in even degrees so the rank equals the Euler characteristic χ(X/T) which again equals |W_{X}| by [48, Prop. 9.5]. By the long exact sequence in cohomology and Nakayama’s lemma we get that H^{∗}(X/T; Z_{p}) is a free Z_{p}- module of rank |W_{X}| as wanted. □

Remark 7.1. For any connected p-compact group X the natural map X/T → BT induces an isomorphism

H_{Q_{p}}^{∗}(X/T) ≅ Q_{p} ⊗_{H_{Qp}^{∗}(BX)} H_{Q_{p}}^{∗}(BT)

since the Eilenberg-Moore spectral sequence of the fibration X/T → BT → BX collapses by [48, Prop. 9.7] and [52, §11]. It then follows from [27] that the natural W_{X}-action on H^{∗}(X/T; Z_{p}) ⊗ Q_{p} = H_{Q_{p}}^{∗}(X/T) is isomorphic to the regular representation of W_{X} when ignoring the grading. Just as for Lie groups this is usually not true over Z_{p} as SU(2)_{2}^{∧} shows.

Proof of Theorem 1.6. By Theorem 1.2 it is enough to prove the statement for the case where X is the p-completion of a compact connected Lie group and the case where X is an exotic p-compact group separately. The case where X is the p-completion of the compact Lie group of course follows directly from the classical Peter-Weyl theorem (cf. e.g. [18, Thm. III.4.1], so we can concentrate on the case where X is exotic. If p does not divide the order of the Weyl group the statement is also obvious: The inclusion T̆ → U(r) induces a map T̆⋊W → U(r|W |) whose p-completion is a faithful representation. The remaining cases have been shown to have faithful representations by Castellana: If (W, L) is in the 2a family then this is carried out in [25] and if (W, L) is one of the pairs (G_{12}, p = 3), (G_{29}, p = 3), (G_{31}, p = 5), or (G_{34}, p = 7) this is carried out in [24]. □

We now turn to Theorem 1.7 which in fact follows easily from the classification. But let us first state the part which one can see by elementary means. (See [52, Lem. 9.3] for the Lie group version.)

Proposition 7.2. Let X be a connected p-compact group then the natural composite map

(L_{X})_{W} ≅ H_{0}(W; H_{2}(BT; Z_{p})) → H_{2}(BX; Z_{p}) ≅ π_{1}(X)

induced by the inclusion T → X is surjective with finite kernel.

In particular if (L_{X})_{W} is torsion free then it is an isomorphism.

Proof. By [86, Thm. 1.4] X is equivalent to a p-compact group of the form (X′ × T′′)/A, where X′ is a simply connected p-compact group, T′′ is a p-compact torus, and A is a finite central subgroup of the product. Since the center of a p-compact group is contained in a maximal torus by [49, Thm. 7.5] we can assume A is a subgroup of T′ × T′′, where T′ is maximal torus for X′, and hence (T′ × T′′)/A is a maximal torus for X. Therefore we get the following diagram of fibration sequences

[figure: BA; BT′ × BT′′; B((T′ × T′′)/A); BA; BX′ × BT′′; BX]

The long exact sequence of homotopy groups and the five-lemma now shows that π_{2}(B((T × T′′)/A)) → π_{2}(BX) is surjective which is the first statement in the proposition. To see that the kernel is finite note that by [48, Thm. 9.7(3)] H_{Q_{p}}^{2}(BX) → H_{Q_{p}}^{2}(BT)^{W} is an isomorphism, which by dualizing to homology shows the claim.

That we get an isomorphism when (L_{X})_{W} is torsion free is obvious from the general statement. □

Remark 7.3. One easily shows that the image of the differential d_{3} : H_{3}(W; Z_{p}) → H_{0}(W; H_{2}(BT; Z_{p})) in the Serre spectral sequence for the fibration BT → BN_{X} → BW is always in the kernel of the surjective map of Proposition 7.2. By standard group cohomology (cf. [26]) the image of this differential identifies with the image of the map given by capping with the k-invariant γ ∈ H^{3}(W; H_{2}(BT; Z_{p})) of the extension. If one knew that

the double coset formula held for p-compact groups (more precisely that H^{∗}(BN; Z_{p}) →^{tr^{∗}} H^{∗}(BX; Z_{p}) →^{res} H^{∗}(BT; Z_{p}) is the identity, cf. [57, Ex. VI.4]) then it would easily follow that this image was in fact equal to the kernel of the map in Proposition 7.2, getting a good proof of the formula for the fundamental group. Note that by a result of Tits [124] (see also [54, 93, 5]) this extension class γ is always of order 2 for compact connected Lie groups. The next proposition gives the complete answer in the Lie case.

Proposition 7.4. Let G be a connected compact Lie group. Then the map π_{1}(T)_{W} → π_{1}(G) is surjective with kernel (Z/2)^{s}, where s is the number of direct factors of G isomorphic to a symplectic group Sp(n), n ≥ 1.

Proof. That the map is surjective follows as in the p-compact case, so we just have to identify the kernel. By [83, Thm. 1.6], for any compact connected Lie group G, T^{W} = Z(G)⊕(Z/2)^{s}, where s is the number of direct factors of G isomorphic to an odd special orthogonal group SO(2n + 1), n ≥ 1.

Consider the dual group G^{∨} of G obtained as the taking the dual root diagram (see [15, §4 8]), then G^{∨} has fundamental group isomorphic to { \atop Z(G)}, where the hat denotes the Poincare dual group (see [15, §4 9]). Likewise { \atop (L_{G}ˇ )_{W}} is canonically isomorphic to T^{W}. Since duality is an involution on the set of compact Lie groups which sends direct summands to direct summands and SO(2n+1) to Sp(n) the claim about the fundamental group follows directly from the dual result about the center. □

Proof of Theorem 1.7. By Theorem 11.1 (L_{X})_{W} = 0 for all the exotic p-compact groups, so Proposition 7.2 shows the formula in this case. By Theorem 1.1 we are hence reduced to showing the formula for X of the form G_{p}^{∧} for some connected compact Lie group G. In this case the formula is well known and easy. Namely it follows from Remark 7.3 that the kernel of (L_{X})_{W} → π_{1}(X) is an elementary abelian 2-group. Alternatively the same conclusion follows from the formula for the fundamental group of a compact Lie group (see [15, §4.6. Prop. 11] or [2, Thm. 5.47], noting that in the notation of [15] (1−s_{θ})v_{θ} = 2v_{θ}). □

We now start to prove Theorem 1.8 and 1.9.

Lemma 7.5. Suppose X and X′ are two connected p-compact groups both with maximal torus normalizer N. Then all elementary abelian p-subgroups of X are toral if and only if all elementary abelian p-subgroups of X′ are toral.

Furthermore, if for all toral elementary abelian p-subgroups V → X the centralizer C_{X}(V) is connected (something which can be calculated from N by [49, Thm. 7.6]) then all elementary abelian p-subgroups in X are toral.

Proof. Suppose that X has a non-toral elementary abelian p-subgroup V → X. We can assume that it is minimal, in the sense that any elementary abelian p-subgroup of smaller rank is toral. Write V = V′ ⊕ V′′, where V′ has rank one. We can assume that V′′ → X factors through T (by the minimality) and that V → X factors through N (by [49, Prop. 2.14]). Let C_{N} (V′′)_{1} denote the maximal torus normalizer of C_{X}(V′′)_{1}, which by [49, Thm. 7.6] can be described in terms of V′′ and N. The adjoint map V′ → C_{N} (V′′) cannot factor through C_{N} (V′′)_{1} since otherwise V′ → C_{X}(V′′) would be conjugate to T in C_{X}(V′′) (by [48, Prop. 5.5]), contradicting that V is assumed not to be toral. Note that CN̆ (V′′)_{1} is normal in CN̆ (V′′) and CN̆ (V)/CN̆ (V)_{1} ≅ π_{0}(C_{X}(V′′)) ≅ π_{0}(C_{X′}(V′′)) (see [49, Rem. 2.11]). Hence V′ → π_{0}(C_{X′}(V′′)) is non-trivial so V → N → X′ cannot be toral in X′.

The last part of the lemma is clear from the proof of the first part. □

Remark 7.6. Despite the above lemma, it is a priori not clear how to determine whether a p-compact group X has the property that all elementary abelian p-subgroups are toral just from looking at N_{X} (but see [121, Thm. 2.28] for the Lie case). However, by a case-by-case analysis (Theorem 1.8), the property is equivalent to that all toral elementary abelian p-subgroups have connected centralizers.

Remark 7.7. Note that by Lannes theory [79, Thm. 0.4] the property that every elementary abelian p-subgroup of X lie inside a maximal torus is equivalent to that H^{∗}(BX; F_{p}) → H^{∗}(BT; F_{p})^{W_{X}} is an F -isomorphism. (See also Theorem 12.1 and Remark 12.3.)

We state the following well known lemma for easy reference:

Lemma 7.8. Suppose that X is a connected p-compact group such that H^{∗}(BX; Z_{p}) is a polynomial algebra with generators concentrated in even degrees. Then all elementary abelian p-subgroups of X are toral.

Proof. For every elementary abelian p-subgroup ν : E → X, H^{∗}(BC_{X}(ν); Z_{p}) is a polynomial algebra with generators concentrated in even degrees by [53, Thm. 1.3] (note that Lannes T -functor preserves objects concentrated in even degrees by [79, Prop. 2.1.3]). In particular C_{X}(ν) is connected so by Lemma 7.5 all elementary abelian p-subgroups are toral. (Alternatively one can use Remark 12.3.) □

Proof of Theorem 1.8. First note that the implication (1) ⇒ (2) follows from Theorem 12.1. The implication (2) ⇒ (3) follows easily from Theorem 5.1. Namely for all toral elementary abelian p-subgroups V → X, Theorem 5.1 implies that W_{C_{X}(V)} is a reflection group, so by [49, Thm. 7.6] C_{X}(V) is connected, using that p is odd. But this implies that all elementary abelian p-subgroups are toral by Lemma 7.5.

We now prove the implication (3) ⇒ (1). First note that by Theorem 11.1 and [50, Thm. 1.4] we can write X ≅ X′ × X′′ where X′ has Weyl group (W_{G}, L_{G} ⊗ Z_{p}), for some compact connected Lie group G, and (W′′, L′′) is a product of exotic finite Z_{p}-reflection groups. Furthermore, since the normalizer of a connected p-compact group is split for p odd by [5, Thm. 1.2] we can in fact choose the compact Lie group such that N_{G_{p}^{∧}} ≅ N_{X′}. Since by Lemma 7.5 the property of having all elementary abelian p-subgroups toral is a property which only depends on N we conclude that G has to have this property as well. But this implies that G has torsion free Z_{p}-cohomology by [7, Thm. B] (see also [121, Thm. 2.28]). In the exotic case, we know by the proof of Theorem 1.4 that we can find a p-compact group X̃′′ which has the same maximal torus normalizer as X′′ and which has torsion free Z_{p}-cohomology.

Hence we have found a p-compact group G_{p}^{∧} × X̃′′ which has the same maximal torus normalizer as X and has torsion free Z_{p}-cohomology. Since by Theorem 1.4 a p-compact group is determined by its normalizer we conclude that X in fact has torsion free Z_{p}- cohomology. (Alternatively, one can appeal to Remark 7.11 which shows that the property of having torsion free Z_{p}-cohomology only depends on N.) □

Remark 7.9. Since the implication (1) ⇒ (3) follows from Lemma 7.8, we see that in the above theorem the implications (1) ⇒ (2) and (1) ⇒ (3) follow by general arguments. In the case of compact Lie groups the implication (3) ⇒ (2) likewise has a general proof, by combining [121, Thm. 2.28] with [36], but at the moment we do not know such a proof for p-compact groups. (See [121, §4].) The remaining implications do not seem to have general proofs even for compact Lie groups.

Proof of Theorem 1.9. As in the proof of Theorem 1.8 we can by Theorem 11.1 and [50, Thm. 1.4] write X ≅ X′ × X′′ where X′ has Weyl group (W_{G}, L_{G} ⊗ Z_{p}), for some compact connected Lie group G, and the Weyl group (W′′, L′′) of X′′ is a product of exotic finite Z_{p}-reflection groups.

By Theorem 1.1 and its proof we know that X′′ is uniquely determined by its Weyl group (W′′, L′′) and has cohomology isomorphic to H^{∗}(B^{2}L′′; Z_{p})^{W′′}. Since H^{2}(B^{2}L′′; Z_{p})^{W′′} = (L′′^{∗})^{W′′} = 0 we conclude by the universal coefficient theorem that π_{1}(X′′) = H_{2}(BX′′; Z_{p}) = 0. Hence by the T -functor, as in the proof of Theorem 1.8, we conclude that X′′ satisfies all three equivalent conditions of the theorem. Furthermore we can, again by [5, Thm. 1.2] since p is odd, choose the compact Lie group such that N_{G_{p}^{∧}} ≅ N_{X′}. Hence by Lemma 7.5 and [49, Thm. 7.6] the theorem holds for X′ if and only if it holds for G. But by [121, Thm. 2.27] the theorem is true for G, which finishes the proof of the theorem. □

Remark 7.10. The same arguments as above shows that the conjectural classification for p = 2 implies that Theorem 1.9 and Theorem 1.8 (1) ⇔ (3) ⇒ (2) holds true for p = 2. However, in Theorem 1.8, (2) is not equivalent to the other conditions since Z_{p}[L_{SO(2n+1)}]^{W} is a polynomial algebra, since this is true for Sp(n), despite SO(2n + 1) having 2-torsion.

Remark 7.11. Notbohm states his classification of connected p-compact groups with Z_{p}[L]^{W} a polynomial algebra in the setup of spaces BX with polynomial cohomology (cf. [97]). This means that his uniqueness statement is a priori only uniqueness amongst pcompact groups with torsion free Z_{p}-cohomology (cf. Theorem 12.1). We will here briefly sketch a direct but case-by-case way (following a line of argument given in a special case in [87, Proof of Thm. 5.3]) to show that for a p-compact the property of having torsion free Z_{p}-cohomology depends only on (W, L), which allows us to remove the extra assumption.

Assume that X is a connected p-compact group, p odd, such that Z_{p}[L_{X}]^{W} is a polynomial algebra. We want to show that H^{∗}(BX; Z_{p}) is a polynomial algebra as well. By Theorem 12.2(1), (L_{X})_{W} is torsion free, so π_{1}(X) = (L_{X})_{W} by Proposition 7.2. By the Serre and Eilenberg-Moore spectral sequences H^{∗}(BX; Z_{p}) is a polynomial algebra if and only if H^{∗}(B(X⟨1⟩); Z_{p}) is a polynomial algebra. Furthermore by construction L_{X⟨1⟩} = SL_{X} and by Theorem 12.2(1) Z_{p}[L_{X⟨1⟩}]^{W_{X}} is also a polynomial algebra, so we can without loss of generality assume that X is simply connected. By [50, Thm. 1.4 and Rem. 1.6] we can furthermore assume that X is a simple p-compact group.

By [5, Thm. 1.2] N̆_{X} = T̆_{X}⋊W. Using the classification of finite Z_{p}-reflection groups (Theorem 11.1) we see that the cohomology of N̆ is always detected on elementary abelian p-subgroups: If p ∤ |W | then the statement is obvious; if (W, L) is in the family 2a then N̆ contains N̆_{U(n)} of coprime index; if (W, L) is either (G_{12}, p = 3), (G_{29}, p = 3), (G_{31}, p = 5), or (G_{34}, p = 7) then it contains N̆_{SU(n)} of coprime index (cf. the proof of Theorem 5.1); if (W, L) is of Lie type then it contains either N̆_{U(n)_{p}^{∧}} or N̆_{SU(n)_{p}^{∧}} of coprime index. Both N̆_{U(n)} and N̆_{SU(n)} both have cohomology which is detected on elementary abelian p-subgroups by [107, Prop. 3.4] (for N_{U(n)_{p}^{∧}}; N_{SU(n)_{p}^{∧}} follows from this). Hence, by a transfer argument, the mod p cohomology of BX is detected on elementary abelian p-subgroups.

Next, we want to show that all elementary abelian p-subgroups of X can be conjugated into a maximal torus. By Lemma 7.5 we just have to show that we can find some p-compact group X′ with the same maximal torus normalizer which has that property. If (W, L) is of Lie type this follows from Borel’s theorem [7, Thm. B]. If (W, L) is exotic this is also true since we know (by Theorem 5.3 or Notbohm’s work [97]) that there exist a p-compact group with Weyl group (W, L) and classifying space having polynomial Z_{p}-cohomology algebra.

The fact that all elementary abelian p-subgroups of X are toral combined with the fact that the cohomology is detected on elementary abelian p-subgroups implies that the mod p cohomology of BX is concentrated in even degrees. Hence H^{∗}(BX; Z_{p}) is torsion free as wanted.

Proof of Theorem 1.10. Let X be a connected finite loop space with maximal torus i : T → X. Note that (X/T)_{p}^{∧} ≃ X_{p}^{∧}/T _{p}^{∧} by the fiber lemma [17, II 5.1], and consequently, by the definition of Euler characteristic, χ(X/T) = χ(X_{p}^{∧}/T _{p}^{∧}). Hence T _{p}^{∧} → X_{p}^{∧} will be a maximal torus normalizer for the p-compact group X_{p}^{∧}, for all primes p.

For our connected finite loop space X, define W_{X}(T) to be the set of conjugacy classes of self-equivalences φ of T such that i and iφ are conjugate. We obviously have an injective homomorphism W_{X} → W_{X_{p}^{∧}} for all primes p and we now want to see that this map is surjective as well, so that we can naturally identify (W_{X}, π_{1}(T) ⊗ Z_{p}) with (W_{X_{p}^{∧}}, L_{X_{p}^{∧}}).

First note that by [48, Proof of Thm. 9.7] we can view W_{X_{p}^{∧}} as the Galois group of the extension of polynomial algebras H_{Q_{p}}^{∗}(BX) → H_{Q_{p}}^{∗}(BT). But, since BX has finitely many cells in each dimension and since BX is nilpotent, we can identify

[figure: H^{∗}(BX; Q) ⊗_{Q} Q_{p}; ≅; H^{∗}(BT; Q) ⊗_{Q} Q_{p}; ≅; H_{Q_{p}}^{∗}(BX); H_{Q_{p}}^{∗}(BT)]

so the extensions H^{∗}(BX; Q) → H^{∗}(BT; Q) and H_{Q_{p}}^{∗}(BX) → H_{Q_{p}}^{∗}(BT) have canonically isomorphic Galois groups. Hence any element in W_{X_{p}^{∧}} lifts to a canonical element in the Galois group of the extension H^{∗}(BX; Q) → H^{∗}(BT; Q). However, since BX_{Q} and BT_{Q} are Eilenberg-Mac Lane spaces (as follows easily by looking at their cohomology), this Galois group just identifies with the self-equivalences BT_{Q} → BT_{Q} which commute with the evaluation map Bi_{Q} : BT_{Q} → BX_{Q} up to homotopy. Hence any element in W_{X_{p}^{∧}} gives rise to a compatible family of self-equivalences of BT _{l}^{∧} and BT _{l}^{∧}, for all primes l. So by the arithmetic square [17, VI.8.1], we get a self-equivalence of BT which commute with evaluation map to BX up to homotopy, i.e., an element in W_{X}. The constructed element is a lift of the element in W_{X_{p}^{∧}} we started with, so the map W_{X} → W_{X_{p}^{∧}} is surjective as well.

Likewise, the argument above showed that W_{X} is the Galois group of the extension H^{∗}(BX; Q) → H^{∗}(BT; Q), so we have an isomorphism

H^{∗}(BX; Q) →^{≅} H^{∗}(BT; Q)^{W_{X}} .

Since H^{∗}(BX; Q) is a polynomial algebra we get by the Shephard-Todd-Chevalley theorem (see [6, Thm. 7.2.1]) that (W_{X}, π_{1}(T)) is a Z-reflection group. Hence, by the proof of Theorem 11.1, (W_{X}, π_{1}(T)) is the Weyl group of some compact connected Lie group G.

For each p we have an extension class x_{p} ∈ H^{3}(W_{X}; π_{1}(T) ⊗ Z_{p}) corresponding to the fibration sequence BT _{p}^{∧} → BN_{X_{p}^{∧}} → BW_{X}. Since H^{3}(W_{X}; π_{1}(T)) is a finite abelian group, and hence given as a sum of its p-primary parts, these extension classes identify with a unique extension class x ∈ H^{3}(W_{X}; π_{1}(T)). We define the loop space N_{X} to be the loop space of the total space in the fibration sequence BT → BN_{X} → BW_{X} with the canonical action of W_{X} on BT and extension class x. Since, the partial F_{p}-completion of BN_{X} identifies with BN_{X_{p}^{∧}}, the arithmetic square produces a canonical morphism N_{X} → X. (N_{X} is, quite naturally, called the maximal torus normalizer of the finite loop space X [87, Def. 1.3].)

By [5] (see also [93], [54]) the extension classes defining T → N_{X} → W_{X} and T → N_{G}(T) → W_{G}(T) are both 2-torsion. Let B̃N denote the fiber-wise Z[\frac{1}{2}]-localization of the total space of the fibration BT → BN_{G}(T) → BW_{G}(T) or equivalently the corresponding fibration with BN_{X}. We hence have embeddings

\xymatrix{ & B̃N \ar[dr] \ar[dl] \\ BX[\frac{1}{2}] &  & BG[\frac{1}{2}]}

By the arithmetic square [17, VI.8.1], the following square is a pullback

[figure: BX[\frac{1}{2}]; ∏p≠2BX_{p}^{∧}; BX_{Q}; (∏_{p≠2} BX_{p}^{∧})_{Q}]

and similarly for BG. By Theorem 1.4 we can construct unique maps between p-completions under B̃N, and we obviously also have a unique map between the rationalizations. By construction (as maps under B̃N) these maps agree on the rationalization of the p-completions, so by the arithmetic square we get an induced map BX[\frac{1}{2}] → BG[\frac{1}{2}], which by construction is an F_{p}-equivalence for all primes p. Since both spaces are one-connected this implies that the map is a homotopy equivalence. □

8. Elementary abelian subgroups of the exceptional groups

In this section we find all conjugacy classes of non-toral elementary abelian p-subgroups E, p odd, of any exceptional compact Lie group G, as well as their centralizers C_{G}(E) and Weyl groups W (E) = N_{G}(E)/C_{G}(E). (Recall that a subgroup of G is called toral if it is contained in a torus in G and non-toral otherwise.) We do this by expanding on the work of Griess [63], who found the maximal non-toral elementary abelian p-subgroups.

Our strategy is as follows. Using the work of Griess [63], we first find representatives of the conjugacy classes of maximal non-toral elementary abelian p-subgroups. We then get lower bounds for their Weyl groups by producing explicit elements in these. From this we are able to identify the non-maximal non-toral elementary abelian p-subgroups and get lower bounds for their Weyl groups. Finally we get exact results on the Weyl groups by computing centralizers.

To be compatible with the standard literature we will in this section state and prove all theorems in the context of linear algebraic groups over the complex numbers C—we state in Proposition 8.4 why this is equivalent to considering compact Lie groups. (The results for G(C) can furthermore be translated into results for G(F) for any algebraically closed field F of characteristic prime to p, see [64, Thm. 1.22] and [60].)

This section is divided into four subsections. The first recalls some results from the theory of linear algebraic groups and discusses the relation to compact Lie groups. The remaining subsections deal with the elementary abelian 3-subgroups of the groups of type E_{6}, E_{7} and E_{8} respectively. (The remaining non-trivial cases E_{8}(C), p = 5 and F_{4}(C), p = 3 are treated completely in [63, Lem. 10.3 and Thm. 7.4].)

For some of our computations for the groups 3E_{6}(C) and E_{8}(C) we have used the computer algebra system Magma [11], although this reliance on computers could if needed be replaced by some rather tedious hand calculations.

Notation 8.1. We use standard names for the linear algebraic groups we consider, i.e. 3E_{6}(C) and E_{6}(C) are respectively the simply connected and the adjoint versions of the group of type E_{6} over C. We let T_{n} denote an n-dimensional torus, i.e. T_{n} = (C^{×})^{n}.

To describe centralizers we follow standard notation for extensions of groups, cf. the ATLAS [32, p. xx]. Thus A : B denotes a group which is the semidirect product of the normal subgroup A with the subgroup B, and A · B denotes a non-split extension of A with B. If p is a prime number, we let p^{n} denote an elementary abelian p-group of rank n.

Whenever E is a concrete elementary abelian p-subgroup of rank r we will always fix an ordered basis of E, so that W (E) may can be considered as a subgroup of GL_{r}(F_{p}). We make the standing convention that all matrices acts on columns.

We identify a permutation σ in the symmetric group Σ_{n} with its permutation matrix A = [a_{ij}] given by a_{ij} = δ_{i,σ(j)} where δ is the Kronecker delta.

If K is a field, we let M_{n}(K) denote set of n × n-matrices over K. For a_{1}, …, a_{n} ∈ K we let diag(a_{1}, …, a_{n}) ∈ M_{n}(K) denote the diagonal matrix with the a_{i}’s in the diagonal. For 1 ≤ i, j ≤ n, we let e_{ij} ∈ M_{n}(K) denotes the matrix whose only non-zero entry is a 1 in position (i, j). Given matrices A_{1} ∈ M_{n_{1}}(K), …, A_{m} ∈ M_{n_{m}}(K) we let A_{1} ⊕ … ⊕ A_{m} denote the n × n-block matrix with the A_{i}’s in the diagonal, n = n_{1} + … + n_{m}. We also need the “blowup” homomorphism Δ_{n,m} : M_{n}(K) ⟶ M_{mn}(K) defined by replacing each entry a_{ij} by a_{ij}I_{m}, where I_{m} ∈ M_{m}(K) is the identity matrix.

As p = 3 in all the cases we consider, we use some special notation. An arbitrary element of F_{3} is denoted by ∗, and ε denotes an element of the multiplicative group F_{3}^{×}. We let ω = e^{2πi/3} and η = e^{2πi/9} and define elements β, γ, τ_{1}, τ_{2} ∈ SL_{3}(C) by β = diag(1, ω, ω^{2}), γ = (1, 2, 3),

τ_{1} = \frac{e^{−πi/18}}{\sqrt{3}} [ \begin{matrix}1 & ω^{2} & 1 \\ 1 & 1 & ω^{2} \\ ω^{2} & 1 & 1 \end{matrix} ]

and τ_{2} = diag(η, η^{−2}, η). Note that β^{τ_{1}} = βγ, γ^{τ_{1}} = γ, β^{τ_{2}} = β and γ^{τ_{2}} = βγ.

8.1. Recollection of some results on linear algebraic groups. Recall that a (not necessarily connected) linear algebraic group G is called reductive if the unipotent radical, i.e., the largest normal connected unipotent subgroup of G, is trivial.

Theorem 8.2. Let G be a linear algebraic group over an algebraically closed field K.

(1) If A is a subgroup of G and S is some subset of A, then A is toral in G if and only if A is toral in C_{G}(S).

(2) If H is a maximal torus of G, then two subsets of H are conjugate in G if and only if they are conjugate in N_{G}(H). If A is a toral subgroup of G, then W (A) = N_{G}(A)/C_{G}(A) is isomorphic to a subquotient of the Weyl group W = N_{G}(H)/H of G.

(3) Assume that G is a connected reductive group such that G′ is simply connected. Then the centralizer of any semisimple element in G is connected. In particular, if A is an abelian subgroup G consisting of semisimple elements generated by at most two elements, then A is toral.

(4) If G is reductive and σ is a semisimple automorphism of G, then G^{σ} is reductive and contains a regular element of G.

(5) Assume that G is a connected reductive group, let Z ⊆ G be a central subgroup, and let π : G → G/Z be the quotient homomorphism. If A is a subgroup of G, then A is toral in G if and only if π(A) is toral in G/Z.

(6) Assume char K = 0 and let 𝔤 be the Lie algebra of G. If S ⊆ G is a finite subset of G, then the Lie algebra of C_{G}(S) is given by

𝔠_{𝔤}(S) = {x ∈ 𝔤| Ad(s)(x) = x for all s ∈ S}.

In particular, if S ⊆ G is a finite subgroup, then

dim C_{G}(S) = \frac{1}{|S|} ∑_{s∈S} tr Ad(s)|_{𝔤}.

Proof. (1): Obviously, if A is toral in C_{G}(S) then A is toral in G. Conversely, if A is toral in G, then A ⊆ H for a torus H in G. Since S ⊆ A we get H ⊆ C_{G}(S) and thus A is toral in C_{G}(S).

(2): The first part follows by a Frattini argument. Assume that A, A^{g} ⊆ H are conjugate subsets of H. Then H and H^{g^{−1}} are maximal tori of C_{G}(A) and thus conjugate in C_{G}(A) (cf. [70, Cor. 21.3.A]). Thus we may write H = H^{g^{−1}c} for some c ∈ C_{G}(A) and we conclude that n = g^{−1}c ∈ N_{G}(H). Then A^{n^{−1}} = A^{c^{−1}g} = A^{g}, which proves the first part. The second part follows similarly, cf. [81, Prop. 1.1(i)].

(3): The first part which is due to Steinberg is proved in [23, Thm. 3.5.6]. The second part follows from the first, cf. [119, II.5.1, p. 206].

(4): We can assume G to be connected. In case G is semisimple and simply connected the first claim is proved in [120, Thm. 8.1] and the general case reduces to this one. Indeed we can find a finite cover G̃ of G which is a direct product of a semisimple simply connected group and a torus, and σ lifts to a semisimple automorphism of G̃ by [120, 9.16]. For the second claim see [130, Thms. 2 and 3] or [119, Pf. of Thm. 5.16,p. E-45] in case G is semisimple; the general case clearly reduces to this one.

(5): By [70, Cor. 21.3.C] we know that if H is a maximal torus of G, then π(H) is a maximal torus of G/Z, and all maximal tori of G/Z are of this form. Thus if A is toral in G, then π(A) is toral in G/Z. Conversely, if H′ is a maximal torus of G/Z containing π(A), then by the above we have H′ = π(H) for some maximal torus H of G. Thus we get A ⊆ ⟨H, Z⟩. However since G is connected and reductive, we get Z ⊆ H by [70, Cor. 26.2.A(b)]. Thus A ⊆ H and we are done.

(6): In case S consists of a single element, the first part follows from [70, Thm. 13.4(a)] (note that the connectivity assumption in [70, Thm. 13.4] is only used in part (b)). The general case follows from this by applying [70, Thm. 12.5(b)] to the centralizers C_{G}(s), s ∈ S.

Now assume that S ⊆ G is a finite subgroup, and let χ denote the character of the adjoint representation of G restricted to S. Then the dimension of

𝔠_{𝔤}(S) = {x ∈ 𝔤| Ad(s)(x) = x for all s ∈ S}.

equals the multiplicity of the trivial character in χ. By the orthogonality relations this is given by

(χ|1) = \frac{1}{|S|} ∑_{s∈S} χ(s),

and we are done. □

We also need the following result whose proof is extracted from [112].

Theorem 8.3. Let G be a reductive linear algebraic group, H a maximal torus of G and let N = N_{G}(H). Let U ⊆ N be a subgroup consisting of semisimple elements such that U/(U ∩ H) is cyclic. Let S be the identity component of H^{U} (the subgroup of H fixed by U), and assume that S is a maximal torus of C_{G}(U). Then C_{N}(U) = N_{C_{G}(U)}(S) and in particular C_{N}(U) is a maximal torus normalizer in C_{G}(U).

Proof. As any element of C_{N}(U) normalizes H^{U} and hence also its identity component S, the inclusion C_{N}(U) ⊆ N_{C_{G}(U)}(S) is clear. Suppose conversely that x ∈ N_{C_{G}(U)}(S). Let C = U ∩ H. From [10, 2.15(d)] it follows that G^{C} is reductive. By assumption the cyclic group U/C acts by semisimple automorphisms on G^{C}. It now follows from Theorem 8.2(4) that G^{U} = (G^{C})^{U/C} is reductive and that every maximal torus of G^{U} is contained in a unique maximal torus of G^{C}. Since C ⊆ H, we see that H is the maximal torus of G^{C} containing S. As H^{x} is also a maximal torus of G^{C} and H^{x} ⊇ S^{x} = S we conclude that H^{x} = H. Thus x ∈ C_{N}(U) proving the result. □

We now explain the relationship between reductive complex linear algebraic groups and compact Lie groups. If G is a complex linear algebraic group then the underlying variety of G is an affine complex variety. By endowing this variety with the usual Euclidean topology instead of the Zariski topology we may view G as a complex Lie group since the group operations are given by polynomial maps.

Proposition 8.4. Let G be a complex linear algebraic group.

(1) Viewed as a Lie group, G contains a maximal compact subgroup which is unique up to conjugacy, and for any such subgroup K we have a diffeomorphism G ≅ K × R^{s} for some s.

(2) Let K be a maximal compact subgroup of G, and let S, S′ ⊆ K be two subsets. If S′ = S^{g} for some g ∈ G, then there exists k ∈ K such that x^{k} = x^{g} for all x ∈ S.

(3) Assume that G is reductive. If S is a finite subgroup of G, then C_{G}(S) is also reductive. If K is a maximal compact subgroup of G containing S, then C_{K}(S) is a maximal compact subgroup of C_{G}(S).

(4) If G is reductive and K is a maximal compact subgroup of G, then we have a diffeomorphism Z(G) ≅ Z(K) × R^{s} for some s.

Proof. Note first that the identity component G_{1} of G seen as a Lie group coincides with the identity component of G seen as a linear algebraic group [105, Ch. 3, §3,no. 1]. Thus G/G_{1} is finite by [70, Prop. 7.3(a)]. The first claim is now part of the Cartan-Chevalley-Iwasawa-Malcev-Mostow theorem [67, Ch. XV, Thm. 3.1] and the second claim also follows from this, cf. [9, Ch. V, §24,Prop. 2].

In case G is reductive it is possible to give a more explicit form of the decomposition above. By [70, Thm. 8.6] we may assume that G is a closed subgroup of GL(V) for some complex vector space V. From [105, Thm. 5.2.8] it follows that G has a compact real form K and we may thus choose a non-degenerate Hermitian inner product on V which is invariant under K (see for instance [105, Thm. 3.4.2]). Let U(V) denote the set of operators in GL(V) which are unitary with respect to the chosen inner product. Using [105, Problems 5.2.3 and 5.2.4] we see that G ⊆ GL(V) is self-adjoint and that K = G ∩ U(V). The last part now follows by combining [105, Cor. 2 of Thm. 5.2.2] with [105, Cor. 2 of Thm. 5.2.1].

If S is a subgroup of K, then S is selfadjoint since K consists of unitary operators. In particular C_{G}(S) is also a selfadjoint subgroup of GL(V), so by [105, Cor. 3 of Thm. 5.2.1] it follows that C_{K}(S) = C_{G}(S) ∩ U(V) is a maximal compact subgroup of C_{G}(S).

It only remains to prove that C_{G}(S) is reductive for a finite subgroup S of G. However by [105, Problem 6.11] and [105, Ch. 4, §1,no. 2] we see that a complex linear algebraic group is reductive if and only if its Lie algebra is reductive. Thus it suffices to prove that the Lie algebra of C_{G}(S) is reductive. However by Theorem 8.2(6) this Lie algebra equals

𝔠_{𝔤}(S) = {x ∈ 𝔤| Ad(s)(x) = x for all s ∈ S}.

where 𝔤 denotes the Lie algebra of G. The claim now follows from [28, Ch. V, §2,no. 2, Prop. 8]. □

8.2. The groups E_{6}(C) and 3E_{6}(C), p = 3. In this subsection we consider the elementary abelian 3-subgroups of the groups of type E_{6} over C. The group 3E_{6}(C) has two nonisomorphic faithful irreducible 27-dimensional representations. These have highest weight λ_{1} and λ_{6} respectively and are dual to each other. An explicit construction of 3E_{6}(C) based on one of these representations was originally given by Freudenthal [58]. This construction is described in more detail in [31, §2] from which we take most of our notation. In particular we let K be the 27-dimensional complex vector space consisting of triples m = (m_{1}, m_{2}, m_{3}) of complex 3 × 3-matrices m_{i}, 1 ≤ i ≤ 3 where addition and scalar multiplication is defined coordinatewise. We define a cubic form ⟨·⟩ on K by

⟨m⟩ = det(m_{1}) + det(m_{2}) + det(m_{3}) − tr(m_{1}m_{2}m_{3}).

Then 3E_{6}(C) is the subgroup of GL(K) preserving the form ⟨·⟩. Moreover the stabilizer in 3E_{6}(C) of the element (I_{3}, 0, 0) ∈ K is the group F_{4}(C). For g_{1}, g_{2}, g_{3} ∈ SL_{3}(C) we have the element s_{g_{1},g_{2},g_{3}} of 3E_{6}(C) given by

s_{g_{1},g_{2},g_{3}} (m_{1}, m_{2}, m_{3}) = (g_{1}m_{1}g_{2}^{−1}, g_{2}m_{2}g_{3}^{−1}, g_{3}m_{3}g_{1}^{−1})

for m = (m_{1}, m_{2}, m_{3}) ∈ K. This gives a representation of SL_{3}(C)^{3} which has kernel C_{3} generated by (ωI_{3}, ωI_{3}, ωI_{3}), and we thus get an embedding of SL_{3}(C)^{3}/C_{3} in 3E_{6}(C). We will denote the element s_{g_{1},g_{2},g_{3}} by [g_{1}, g_{2}, g_{3}].

We let {e_{j,k}^{i}}, 1 ≤ i, j, k ≤ 3 be the natural basis of K consisting of the elements e_{j,k}^{i} whose entries are all 0 except for the (j, k)-entry of the i’th matrix which equals 1. The elements of 3E_{6}(C) which acts diagonally with respect to this basis of K form a maximal torus H in 3E_{6}(C). Let m_{i}^{j,k} denote the (j, k)-entry of the matrix m_{i}. We then have H-invariant subgroups

u_{α_{1}}(t) = [I_{3}, I_{3} + te_{1,3}, I_{3}] , u_{−α_{1}}(t) = [I_{3}, I_{3} + te_{3,1}, I_{3}] ,
u_{α_{2}}(t) = [I_{3} + te_{2,1}, I_{3}, I_{3}] , u_{−α_{2}}(t) = [I_{3} + te_{1,2}, I_{3}, I_{3}] ,
u_{α_{3}}(t) = [I_{3}, I_{3} + te_{2,1}, I_{3}] , u_{−α_{3}}(t) = [I_{3}, I_{3} + te_{1,2}, I_{3}] ,

u_{α_{4}}(t) : (m_{i})_{i=1,2,3} ↦ (m_{i} + t · [ \begin{matrix}0 & −m_{i+2}^{2,3} & 0 \\ 0 & 0 & 0 \\ 0 & m_{i+2}^{2,1} & 0 \end{matrix} ])_{i=1,2,3}

u_{−α_{4}}(t) : (m_{i})_{i=1,2,3} ↦ (m_{i} + t · [ \begin{matrix}0 & 0 & 0 \\ m_{i+1}^{3,2} & 0 & −m_{i+1}^{1,2} \\ 0 & 0 & 0 \end{matrix} ])_{i=1,2,3}

u_{α_{5}}(t) = [I_{3}, I_{3}, I_{3} + te_{2,1}] , u_{−α_{5}}(t) = [I_{3}, I_{3}, I_{3} + te_{1,2}] ,
u_{α_{6}}(t) = [I_{3}, I_{3}, I_{3} + te_{1,3}] , u_{−α_{6}}(t) = [I_{3}, I_{3}, I_{3} + te_{3,1}] .

Here, in the description of u_{±α_{4}}(t), the m_{i}’s should be counted cyclicly mod 3, e.g. m_{i+2} = m_{1} for i = 2.

The associated roots α_{i}, 1 ≤ i ≤ 6, of these root subgroups form a base for the root system Φ(E_{6}) of 3E_{6}(C) (our numbering agrees with [13, p. 260–262]). For this base of Φ(E_{6}), the highest weight of K is λ_{1}. Furthermore the root subgroups u_{±α_{i}}, 1 ≤ i ≤ 6, have been chosen so that they satisfy [118, 8.1.1(i) and 8.1.4(i)], i.e. they form part of a realization ([118, p. 133]) of Φ(E_{6}) in 3E_{6}(C). For α = ±α_{i}, 1 ≤ i ≤ 6, and t ∈ C^{×}, we may then define the elements

n_{α}(t) = u_{α}(t)u_{−α}(−1/t)u_{α}(t), h_{α}(t) = n_{α}(t)n_{α}(1)^{−1}.

Then the maximal torus consists of the elements h(t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}) = ∏_{i=1}^{6} h_{α_{i}}(t_{i}) and the normalizer N(H) of the maximal torus is generated by H and the elements n_{i} = n_{α_{i}}(1), 1 ≤ i ≤ 6. It should be noted that this notation differs from the one used in [31]. More precisely, the element h(α, β, γ, δ, ϵ, ζ) in [31] is h(δ, α^{−1}, γ^{−1}, β, ϵ^{−1}, ζ) in our notation, and the elements n_{1}, n_{2}, n_{3}, n_{4}, n_{5} and n_{6} in [31] equals respectively n_{1}h_{α_{1}}(−1)h_{α_{3}}(−1), n_{2}h(−1, 1, 1, −1, 1, −1), n_{3}h_{α_{1}}(−1), n_{4}, n_{5}h_{α_{6}}(−1) and n_{6}h_{α_{5}}(−1)h_{α_{6}}(−1) in our notation.

From the description of the root system of type E_{6} in [13, p. 260–262] we see that the center Z of 3E_{6}(C) is cyclic of order 3 and is generated by the element z = [I_{3}, ω^{2}I_{3}, ωI_{3}]. We consider also the element a = [ωI_{3}, I_{3}, I_{3}]. A straightforward computation shows that the roots of the centralizer C_{3E_{6}(C)}(a) are

{±α_{1}, ±α_{2}, ±α_{3}, ±α_{5}, ±α_{6}, ±α̃, ±(α_{1} + α_{3}), ±(α_{5} + α_{6}), ±(α_{2} − α̃)},

where α̃ is the longest root. The Dynkin diagram for this centralizer is the same as the extended Dynkin diagram for E_{6} with the node α_{4} removed. In particular it has type A_{2}A_{2}A_{2} and basis {α_{1}, α_{3}, α_{5}, α_{6}, α_{2}, −α̃}. Since 3E_{6}(C) is simply connected, Theorem 8.2(3) implies that the centralizer C_{3E_{6}(C)}(a) is connected, and thus it is generated by the maximal torus H and the root subgroups u_{±α}(t) where α runs through the simple roots in the basis {α_{1}, α_{3}, α_{5}, α_{6}, α_{2}, −α̃} of the root system of centralizer. Now note that uα̃(t) = [I_{3} + te_{3,1}, I_{3}, I_{3}] and u_{−α̃}(t) = [I_{3} + te_{1,3}, I_{3}, I_{3}] are root subgroups with associated roots α̃ and −α̃ respectively. Since these along with H and the root subgroups u_{±α_{1}}, u_{±α_{2}}, u_{±α_{3}}, u_{±α_{5}} and u_{±α_{6}} generate the subgroup SL_{3}(C)^{3}/C_{3} of 3E_{6}(C) from above, we conclude that C_{3E_{6}(C)}(a) = SL_{3}(C)^{3}/C_{3}.

To describe the conjugacy classes of elementary abelian 3-subgroups we need to introduce some more elements. Consider the following elements in SL_{3}(C)^{3}/C_{3} ⊆ 3E_{6}(C):

x_{1} = [I_{3}, β, β] ,  x_{2} = [β, β, β] ,  y_{1} = [I_{3}, γ, γ^{2}] ,  y_{2} = [γ, γ, γ] .

We also need the following elements in N(H):

s_{1} = n_{1}n_{3}n_{4}n_{2}n_{5}n_{4}n_{3}n_{1}n_{6}n_{5}n_{4}n_{2}n_{3}n_{4}n_{5}n_{6},
s_{2} = n_{1}n_{2}n_{3}n_{1}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{2}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}·
n_{3}n_{5}n_{4}n_{2}n_{6}n_{5}n_{4}n_{3}n_{1}

The action of these elements are as follows:

s_{1}(m_{1}, m_{2}, m_{3}) = (m_{3}, m_{1}, m_{2}), s_{2}(m_{1}, m_{2}, m_{3}) = (m_{3}^{T}, m_{2}^{T}, m_{1}^{T}),

where m_{i}^{T} denotes the transpose of m_{i}. Thus these elements acts by conjugation on the subgroup SL_{3}(C)^{3}/C_{3} as follows

[g_{1}, g_{2}, g_{3}]^{s_{1}} = [g_{2}, g_{3}, g_{1}] , [g_{1}, g_{2}, g_{3}]^{s_{2}} = [(g_{1}^{−1})^{T} , (g_{3}^{−1})^{T} , (g_{2}^{−1})^{T}] .

Lemma 8.5. We have

z = h(ω, 1, ω^{2}, 1, ω, ω^{2}), a = h(ω, 1, ω^{2}, 1, ω^{2}, ω), x_{1} = h(ω, 1, ω, 1, ω, ω),
x_{2} = h(1, ω^{2}, ω^{2}, 1, ω^{2}, 1), y_{1} = n_{1}n_{3}n_{5}n_{6}h_{α_{5}}(−1),
y_{2} = n_{1}n_{2}n_{3}n_{4}n_{3}n_{1}n_{5}n_{4}n_{2}n_{3}n_{4}n_{5}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{6}n_{5}h_{α_{2}}(−1).

Moreover conjugation by the element

n_{1}n_{4}n_{2}n_{3}n_{1}n_{4}n_{5}n_{4}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4} · h_{α_{2}}(−1)h_{α_{4}}(−1)

acts as follows:

a ↦ x_{2},  x_{2} ↦ a,  y_{1} ↦ s_{1},  y_{2} ↦ y_{2}^{2},  x_{2}x_{1}^{−1} ↦ h_{α_{4}}(ω) = [τ_{2}, τ_{2}, τ_{2}] .

Proof. Both parts of the lemma may be checked by direct computation. The second part also follows from the first by using the following relations in N(H): The element n_{i} has image s_{α_{i}} in W ([118, 8.1.4(i)]), we have n_{i}^{2} = h_{α_{i}}(−1) ([118, 8.1.4(ii)]) and

n_{i}n_{j}n_{i} … = n_{j}n_{i}n_{j} …

for 1 ≤ i, j ≤ 6, where the number of factors on both sides equals the order of s_{α_{i}}s_{α_{j}} in W ([118, 9.3.2]). □

Notation 8.6. For our calculations, we need some information on the conjugacy classes of elements of order 3 in 3E_{6}(C). These are given in [31, Table 2]: There are 7 such conjugacy classes, which we label 3A, 3B, 3B′, 3C, 3D, 3E and 3E′, where 3B′ and 3E′ denotes the inverses of the classes 3B and 3E. This notation is almost identical to the notation in [31], but differs from [63]. We will need the following, which follows quickly from [31, Table 2] using the action of W on H: We have z ∈ 3E, a, x_{2}, y_{2} ∈ 3C, x_{1}, y_{1} ∈ 3D and x_{2}x_{1}^{−1} ∈ 3A. Multiplication by z acts as follows on the conjugacy classes:

3A ↦ 3B, 3B ↦ 3B′, 3B′ ↦ 3A, 3C ↦ 3C, 3D ↦ 3D, 3E ↦ 3E′, 3E′ ↦ 1,

where 1 denotes the conjugacy class consisting of the identity element.

Theorem 8.7. The conjugacy classes of non-toral elementary abelian 3-subgroups of 3E_{6}(C) are given by the following table.

rank name

ordered basis 3E_{6}(C)-class distribution C_{3E_{6}(C)}(E)
3 E_{3E_{6}}^{3}  ⟨a, x_{2}, y_{2}⟩  3C^{26}  E_{3E_{6}}^{4}
4 E_{3E_{6}}^{4} ⟨z, a, x_{2}, y_{2}⟩  3C^{78}3E^{1}3E′^{1}  E_{3E_{6}}^{4}

Their Weyl groups with respect to the given ordered bases are as follows:

W (E_{3E_{6}}^{3}) = SL_{3}(F_{3}), W (E_{3E_{6}}^{4}) = [ \begin{matrix}1 & ∗ ∗ ∗ \\ 0 0 0 & SL_{3}(F_{3}) \end{matrix} ]

Proof. Non-toral subgroups: By [63, Thm. 11.13], there are two conjugacy classes of nontoral elementary abelian 3-subgroups in 3E_{6}(C), one non-maximal of rank three and one maximal of rank 4. We may concretely realize these as follows. Consider the subgroups

E_{3E_{6}}^{3} = ⟨a, x_{2}, y_{2}⟩ and E_{3E_{6}}^{4} = ⟨z, a, x_{2}, y_{2}⟩ ,

which are readily seen to be elementary abelian 3-subgroups of rank 3 and 4 respectively. In particular both groups are subsets of C_{3E_{6}(C)}(a) = SL_{3}(C)^{3}/C_{3}, and since β, γ ∈ SL_{3}(C) does not commute, we see that the preimages of E_{3E_{6}}^{3} and E_{3E_{6}}^{4} under the projection SL_{3}(C)^{3} → SL_{3}(C)^{3}/C_{3} are non-abelian. Thus by Theorem 8.2(5) E_{3E_{6}}^{3} and E_{3E_{6}}^{4} are nontoral in SL_{3}(C)^{3}/C_{3} = C_{3E_{6}(C)}(a) and hence also non-toral in 3E_{6}(C) by Theorem 8.2(1). Thus by the above these two groups represent the conjugacy classes of non-toral elementary abelian 3-subgroups in 3E_{6}(C).

Lower bounds for Weyl groups: By [63, Thm. 7.4] there is a unique non-toral elementary abelian 3-subgroup of F_{4}(C) of rank 3. Since we have an inclusion F_{4}(C) ⊆ 3E_{6}(C) this subgroup may also be considered as a subgroup of 3E_{6}(C). As its Weyl group in F_{4}(C) is SL_{3}(F_{3}), its Weyl group in 3E_{6}(C) must contain SL_{3}(F_{3}). In particular it has order divisible by 13 and since the order of W (E_{6}) is 2^{7} · 3^{4} · 5 which is not divisible by 13, we conclude by Theorem 8.2(2) that E is non-toral in 3E_{6}(C) as well. Thus by the above E must be conjugate to E_{3E_{6}}^{3}, and we get that W (E_{3E_{6}}^{3}) contains SL_{3}(F_{3}). From this we immediately see that W (E_{3E_{6}}^{4}) contains the group

[ \begin{matrix}1 & 0 0 0 \\ 0 0 0 & SL_{3}(F_{3}) \end{matrix} ]

Note that the element [I_{3}, β, β^{2}] commutes with z, a and x_{2} and conjugates y_{2} to y_{2}z. Thus it normalizes E_{3E_{6}}^{4} and produces the element I_{4} + e_{1,4} in W (E_{3E_{6}}^{4}). As a result we see that W (E_{3E_{6}}^{4}) contains the group

[ \begin{matrix}1 & ∗ ∗ ∗ \\ 0 0 0 & SL_{3}(F_{3}) \end{matrix} ]

Class distributions: Since a ∈ 3C by 8.6 and W (E_{3E_{6}}^{3}) contains SL_{3}(F_{3}) which acts transitively on E_{3E_{6}}^{3} \ {1}, the class distribution of E_{3E_{6}}^{3} follows immediately. Using this and the information given in 8.6 about the action of z on the conjugacy classes, the class distribution of E_{3E_{6}}^{4} is easily found.

Centralizers: We have already seen that C_{3E_{6}(C)}(a) = SL_{3}(C)^{3}/C_{3}. From this we directly get

C_{3E_{6}(C)}(a, x_{2}) = C_{SL_{3}(C)^{3}/C_{3}}(x_{2}) = ⟨y_{2}, (T_{2} × T_{2} × T_{2}) /C_{3}⟩ ,
C_{3E_{6}(C)}(a, x_{2}, y_{2}) = ⟨x_{2}, y_{2}, (⟨ωI_{3}⟩ × ⟨ωI_{3}⟩ × ⟨ωI_{3}⟩) /C_{3}⟩ = E_{3E_{6}}^{4},

proving that C_{3E_{6}(C)}(E_{3E_{6}}^{3}) = C_{3E_{6}(C)}(E_{3E_{6}}^{4}) = E_{3E_{6}}^{4}.

Exact Weyl groups: From the lower bounds above and the fact that z is central we get SL_{3}(F_{3}) ⊆ W (E_{3E_{6}}^{3}) ⊆ GL_{3}(F_{3}) and

[ \begin{matrix}1 & ∗ ∗ ∗ \\ 0 0 0 & SL_{3}(F_{3}) \end{matrix} ] ⊆ W (E_{3E_{6}}^{4}) ⊆ [ \begin{matrix}1 & ∗ ∗ ∗ \\ 0 0 0 & GL_{3}(F_{3}) \end{matrix} ]

As C_{3E_{6}(C)}(a, x_{2}) = ⟨y_{2}, (T_{2} × T_{2} × T_{2}) /C_{3}⟩, we see that no element in C_{3E_{6}(C)}(a, x_{2}) conjugates y_{2} to y_{2}^{−1}. Hence diag(1, 1, 2) ∉ W (E_{3E_{6}}^{3}) and diag(1, 1, 1, 2) ∉ W (E_{3E_{6}}^{4}) which shows that Weyl groups are the ones given in the Theorem. □

We now turn to the group E_{6}(C). As above we let Z be the center of 3E_{6}(C) and we let π : 3E_{6}(C) → E_{6}(C) = 3E_{6}(C)/Z denote the projection. For g ∈ 3E_{6}(C) we write \overline{g} instead of π(g) and similarly we let \overline{S} = π(S) for a subset S ⊆ 3E_{6}(C).

Lemma 8.8. Let E be a rank 2 non-toral elementary abelian 3-subgroup of E_{6}(C). Then the Weyl group W (E) is a subgroup of SL_{2}(F_{3}).

Proof. Let E = ⟨\overline{g_{1}}, \overline{g_{2}}⟩. By Theorem 8.2 part (5) and (3) the group ⟨g_{1}, g_{2}⟩ ⊆ 3E_{6}(C) is non-abelian. Thus setting z′ = [g_{1}, g_{2}] ∈ Z we have z′ ≠ 1. Assume that σ ∈ W (E) is

represented by the matrix [ \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} ], i.e. we have σ(\overline{g_{1}}) = (\overline{g_{1}})^{a_{11}}(\overline{g_{2}})^{a_{21}} and σ(\overline{g_{2}}) =

(\overline{g_{1}})^{a_{12}}(\overline{g_{2}})^{a_{22}}. Since σ is given by a conjugation in E_{6}(C), it lifts to a conjugation in 3E_{6}(C). Now the relation [g_{1}, g_{2}] = z′ ∈ Z shows (z′)^{a_{11}·a_{22}−a_{12}·a_{21}} = z′, so since z′ ≠ 1 we have σ ∈ SL_{2}(F_{3}). □

Theorem 8.9. The conjugacy classes of non-toral elementary abelian 3-subgroups of E_{6}(C) are given by the following table:

rank name

ordered basis  3E_{6}(C)-class distribution  C_{E_{6}(C)}(E)  Z(C_{E_{6}(C)}(E))
2  E_{E_{6}}^{2a}  ⟨\overline{y_{1}}, \overline{x_{2}}⟩  3C^{18}3D^{6}3E^{1}3E′^{1}  E_{E_{6}}^{2a} × PSL_{3}(C)  E_{E_{6}}^{2a}
2  E_{E_{6}}^{2b}  ⟨\overline{y_{1}}, \overline{x_{1}}⟩  3D^{24}3E^{1}3E′^{1}  E_{E_{6}}^{2b} × G_{2}(C)  E_{E_{6}}^{2b}
3  E_{E_{6}}^{3a}  ⟨\overline{a}, \overline{y_{1}}, \overline{x_{2}}⟩  3C^{60}3D^{18}3E^{1}3E′^{1}  E_{E_{6}}^{3a} ∘_{⟨\overline{a}⟩} (T_{2} : ⟨\overline{y_{2}}⟩)  E_{E_{6}}^{3a}
3  E_{E_{6}}^{3b}  ⟨\overline{a}, \overline{x_{2}}, \overline{y_{2}}⟩  3C^{78}3E^{1}3E′^{1}  E_{E_{6}}^{3b} · 3^{3}  E_{E_{6}}^{3b}
3  E_{E_{6}}^{3c}  ⟨\overline{a}, \overline{y_{1}}, \overline{x_{1}}⟩  3C^{6}3D^{72}3E^{1}3E′^{1}  E_{E_{6}}^{3c} ∘_{⟨\overline{a}⟩} SL_{3}(C)  E_{E_{6}}^{3c}
3  E_{E_{6}}^{3d}  ⟨\overline{x_{2}x_{1}^{−1}}, \overline{y_{1}}, \overline{x_{1}}⟩ 3A^{2}3B^{2}3B′^{2}3C^{48}3D^{24}3E^{1}3E′^{1} E_{E_{6}}^{3d} ∘_{⟨\overline{x_{2}x_{1}^{−1}}⟩} GL_{2}(C) E_{E_{6}}^{3d} ∘_{⟨\overline{x_{2}x_{1}^{−1}}⟩} T_{1}
4  E_{E_{6}}^{4a}  ⟨\overline{a}, \overline{y_{2}}, \overline{y_{1}}, \overline{x_{2}}⟩  3C^{186}3D^{54}3E^{1}3E′^{1}  E_{E_{6}}^{4a}  E_{E_{6}}^{4a}
4  E_{E_{6}}^{4b}  ⟨\overline{a}, \overline{x_{2}x_{1}^{−1}}, \overline{y_{1}}, \overline{x_{1}}⟩ 3A^{6}3B^{6}3B′^{6}3C^{150}3D^{72}3E^{1}3E′^{1} E_{E_{6}}^{4b} ∘_{⟨\overline{a},\overline{x_{2}x_{1}^{−1}}⟩} T_{2} E_{E_{6}}^{4b} ∘_{⟨\overline{a},\overline{x_{2}x_{1}^{−1}}⟩} T_{2}

In particular we have _{3}Z(C_{E_{6}(C)}(E)) = E for any non-toral elementary abelian 3-subgroup of E_{6}(C). (In the table the 3E_{6}(C)-class distribution of E ⊆ E_{6}(C) denotes the class distribution of π^{−1}(E) ⊆ 3E_{6}(C).)

The Weyl groups of these groups with respect to the given ordered bases are given as follows:

W (E_{E_{6}}^{2a}) = [ \begin{matrix}ε & ∗ \\ 0 & ε \end{matrix} ] , W (E_{E_{6}}^{2b}) = SL_{2}(F_{3}), W (E_{E_{6}}^{3a}) = [ \begin{matrix}ε_{1} & ∗ & ∗ \\ 0 & ε_{2} & ∗ \\ 0 & 0 & ε_{2} \end{matrix} ]

W (E_{E_{6}}^{3b}) = SL_{3}(F_{3}), W (E_{E_{6}}^{3c}) = [ \begin{matrix}ε & ∗ ∗ \\ {0 \atop 0} & SL_{2}(F_{3}) \end{matrix} ] , W (E_{E_{6}}^{3d}) = [ \begin{matrix}ε & 0 0 \\ {0 \atop 0} & SL_{2}(F_{3}) \end{matrix} ]

W (E_{E_{6}}^{4a}) = [ \begin{matrix}GL_{2}(F_{3}) & \begin{matrix}∗ \\ ∗ \end{matrix} & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 0 & det & ∗ \\ 0 0 & 0 & det \end{matrix} ] , W (E_{E_{6}}^{4b}) = [ \begin{matrix}ε_{1} & ∗ & ∗ ∗ \\ 0 & ε_{2} & 0 0 \\ {0 \atop 0} & {0 \atop 0} & SL_{2}(F_{3}) \end{matrix} ]

where det denotes the determinant of the matrix from GL_{2}(F_{3}) in the description of W (E_{E_{6}}^{4a}).

Proof. Maximal non-toral subgroups: By [63, Them. 11.14], there are two conjugacy classes of maximal non-toral elementary abelian 3-subgroups in E_{6}(C), both of which have rank 4.

We may concretely realize these as follows. Consider the subgroups

E_{a} = ⟨z, a, y_{1}, y_{2}, x_{2}⟩ and E_{b} = ⟨z, a, x_{2}x_{1}^{−1}, y_{1}, x_{1}⟩

of C_{3E_{6}(C)}(a) = SL_{3}(C)^{3}/C_{3}. Since the commutator subgroup of both of these is Z, we see that E_{E_{6}}^{4a} = π(E_{a}) and E_{E_{6}}^{4b} = π(E_{b}) are elementary abelian subgroups of rank 4 in E_{6}(C). It follows from Theorem 8.2(5) that both E_{E_{6}}^{4a} and E_{E_{6}}^{4b} are non-toral in E_{6}(C). We will see below that their class distributions are as given in the table. From this it follows that they are not conjugate and thus represents the two conjugacy classes of maximal elementary abelian 3-subgroups in E_{6}(C).

Lower bounds for Weyl groups of maximal non-toral subgroups: We can find lower bounds for the Weyl groups of the maximal non-toral elementary abelian 3-subgroups by conjugating with elements coming from the centralizer C_{3E_{6}(C)}(a) = SL_{3}(C)^{3}/C_{3} and the normalizer N(H) of the maximal torus.

The elements \overline{[β^{2}, I_{3}, I_{3}]}, \overline{[I_{3}, τ_{1}, τ_{1}^{2}]}, \overline{s_{1}} and \overline{s_{2}} normalize E_{E_{6}}^{4a} and conjugation by these elements induce the automorphisms given by the matrices I_{4} + e_{1,2}, I_{4} + e_{3,4}, I_{4} + e_{2,3} and diag(2, 1, 2, 2) on E_{E_{6}}^{4a}. Moreover, by Lemma 8.5 we may conjugate the ordered basis of E_{E_{6}}^{4a}

into the ordered basis ⟨\overline{x_{2}}, \overline{y_{2}^{2}}, \overline{s_{1}}, \overline{a}⟩ . Noting that the element \overline{[τ_{1}, τ_{1}, τ_{1}]} commutes with \overline{y_{2}},

\overline{s_{1}} and \overline{a} and conjugates \overline{x_{2}} into \overline{x_{2}y_{2}}, we see that W (E_{E_{6}}^{4a}) contains the element I_{4} + e_{2,1}. The above matrices are easily seen to generate the group

W′(E_{E_{6}}^{4a}) = [ \begin{matrix}GL_{2}(F_{3}) & \begin{matrix}∗ \\ ∗ \end{matrix} & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 0 & det & ∗ \\ 0 0 & 0 & det \end{matrix} ]

and thus W_{E_{6}}^{4a} contains this group.

Now consider E_{E_{6}}^{4b} and let σ = −(2, 3) ∈ SL_{3}(C). We then see that the elements \overline{[I_{3}, τ_{1}, τ_{1}^{2}]}, \overline{[I_{3}, τ_{2}β, τ_{2}^{2}]}, \overline{[σ, I_{3}, I_{3}]}, \overline{[γ, I_{3}, I_{3}]}, \overline{[I_{3}, β^{2}, I_{3}]} and \overline{s_{2}} normalize E_{E_{6}}^{4b}, and conjugation by these elements induce the automorphisms given by the matrices I_{4} + e_{3,4}, I_{4} + e_{4,3}, diag(1, 2, 1, 1), I_{4} + e_{1,2}, I_{4} + e_{1,3} and −I_{4} on E_{E_{6}}^{4b}. These matrices generate the group

W′(E_{E_{6}}^{4b}) = [ \begin{matrix}ε_{1} & ∗ & ∗ ∗ \\ 0 & ε_{2} & 0 0 \\ {0 \atop 0} & {0 \atop 0} & SL_{2}(F_{3}) \end{matrix} ]

and thus W_{E_{6}}^{4b} contains this group.

Orbit computation: Any elementary abelian 3-subgroup of rank 1 is toral since E_{6}(C) is connected. As we already know that E_{E_{6}}^{4a} and E_{E_{6}}^{4b} are representatives of the maximal non-toral elementary abelian 3-subgroups, we may find the conjugacy classes of non-toral elementary abelian 3-subgroups of rank 2 and 3 by studying subgroups of these.

Under the action of W′(E_{E_{6}}^{4a}), the set of rank 2 subgroups of E_{E_{6}}^{4a} has orbit representatives

E_{E_{6}}^{2a} = ⟨\overline{y_{1}}, \overline{x_{2}}⟩ , ⟨\overline{a}, \overline{x_{2}}⟩ , ⟨\overline{a}, \overline{y_{1}}⟩ and ⟨\overline{a}, \overline{y_{2}}⟩ ,

and under the action of W′(E_{E_{6}}^{4b}), the set of rank 2 subgroups of E_{E_{6}}^{4b} has orbit representatives

E_{E_{6}}^{2a} = ⟨\overline{y_{1}}, \overline{x_{2}}⟩ , E_{E_{6}}^{2b} = ⟨\overline{y_{1}}, \overline{x_{1}}⟩ , ⟨\overline{a}, \overline{x_{2}}⟩ , ⟨\overline{a}, \overline{y_{1}}⟩ , ⟨\overline{a}, \overline{x_{2}x_{1}^{−1}}⟩ and ⟨\overline{x_{2}x_{1}^{−1}}, \overline{x_{1}}⟩ .

Similarly we find that under the action of W′(E_{E_{6}}^{4a}), the set of rank 3 subgroups of E_{E_{6}}^{4a} has orbit representatives

E_{E_{6}}^{3a} = ⟨\overline{a}, \overline{y_{1}}, \overline{x_{2}}⟩ , E_{E_{6}}^{3b} = ⟨\overline{a}, \overline{y_{2}}, \overline{x_{2}}⟩ and ⟨\overline{a}, \overline{y_{1}}, \overline{y_{2}}⟩ ,

and that under the action of W′(E_{E_{6}}^{4b}), the set of rank 3 subgroups of E_{E_{6}}^{4b} has orbit representatives

E_{E_{6}}^{3a} = ⟨\overline{a}, \overline{y_{1}}, \overline{x_{2}}⟩ , E_{E_{6}}^{3c} = ⟨\overline{a}, \overline{y_{1}}, \overline{x_{1}}⟩ , E_{E_{6}}^{3d} = ⟨\overline{x_{2}x_{1}^{−1}}, \overline{y_{1}}, \overline{x_{1}}⟩ and ⟨\overline{a}, \overline{x_{2}x_{1}^{−1}}, \overline{x_{1}}⟩ .

Other non-toral subgroups: We see directly that the subgroups ⟨\overline{a}, \overline{x_{2}}⟩, ⟨\overline{a}, \overline{x_{2}x_{1}^{−1}}⟩, ⟨\overline{x_{2}x_{1}^{−1}}, \overline{x_{1}}⟩

and ⟨\overline{a}, \overline{x_{2}x_{1}^{−1}}, \overline{x_{1}}⟩ are toral. Noting that the elements β and γ are conjugate in SL_{3}(C) we

see that the group ⟨\overline{a}, \overline{y_{1}}, \overline{y_{2}}⟩ is conjugate to the group ⟨\overline{a}, \overline{[I_{3}, β, β^{2}]}, \overline{x_{2}}⟩ which is obviously toral. Thus we see that the groups ⟨\overline{a}, \overline{y_{1}}, \overline{y_{2}}⟩, ⟨\overline{a}, \overline{y_{1}}⟩ and ⟨\overline{a}, \overline{y_{2}}⟩ are also toral. Using the fact that [y_{1}, x_{1}] = [y_{1}, x_{2}] = z we see from Theorem 8.2(5) that both E_{E_{6}}^{2a} and E_{E_{6}}^{2b} are non-toral in E_{6}(C). Since the groups E_{E_{6}}^{3a}, E_{E_{6}}^{3c} and E_{E_{6}}^{3d} all contain either E_{E_{6}}^{2a} or E_{E_{6}}^{2b} they are also non-toral. Using Theorem 8.2(5) we see that the group E_{E_{6}}^{3b} is non-toral in E_{6}(C), since we know that π^{−1}(E_{E_{6}}^{3b}) = E_{3E_{6}}^{4} is non-toral in 3E_{6}(C) by Theorem 8.7.

Class distributions: Using 8.6 and the action of the groups W′(E_{E_{6}}^{4a}) and W′(E_{E_{6}}^{4b}) it is not hard to verify the class distributions in the table. As an example consider the group E_{E_{6}}^{4b}. From the action of W′(E_{E_{6}}^{4b}) we see that E_{E_{6}}^{4b} \ {1} contains 2 elements conjugate to

\overline{a}, 6 elements conjugate to \overline{x_{2}x_{1}^{−1}}, 24 elements conjugate to \overline{x_{1}} and 48 elements conjugate to \overline{x_{2}}. Thus by 8.6, the set π^{−1}(E_{E_{6}}^{4b} \ \{1\}) contains 6 elements from each of the classes 3A, 3B and 3B′, 3 · (2 + 48) = 150 elements from the class 3C and 3 · 24 = 72 elements from the class 3D. Including the elements z and z^{2} from the classes 3E and 3E′ respectively, we get the class distribution of π^{−1}(E_{E_{6}}^{4b}) \ {1} given in the table. Similar computations give the remaining entries in the table. Since these distributions are different we see that the groups in the table are not conjugate and thus they provide a set of representatives for the conjugacy classes of non-toral elementary abelian 3-subgroups of E_{6}(C).

Lower bounds for other Weyl groups: We now show that the other matrix groups in the table are all lower bounds for the remaining Weyl groups. To do this consider one of the non-maximal groups E from the table. We then have E ⊆ E_{E_{6}}^{4a} or E ⊆ E_{E_{6}}^{4b}, and we get a lower bound on W (E) by considering the action on E of the subgroup of W′(E_{E_{6}}^{4a}) or W′(E_{E_{6}}^{4b}) fixing E. As an example we see that E_{E_{6}}^{2a} ⊆ E_{E_{6}}^{4a} and that the stabilizer of E_{E_{6}}^{2a} inside W′(E_{E_{6}}^{4a}) is

GL_{2}(F_{3}) {0 \atop 0} {0 \atop 0}
0 0 det x
0 0  0 det

where det is the determinant of the matrix from GL_{2}(F_{3}). The action of such a matrix on E_{E_{6}}^{2a} is given by

\overline{y_{1}} ↦ (\overline{y_{1}})det,  \overline{x_{2}} ↦ (\overline{y_{1}})^{x}(\overline{x_{2}})det.

Thus W (E_{E_{6}}^{2a}) contains the group

W′(E_{E_{6}}^{2a}) = [ \begin{matrix}ε & ∗ \\ 0 & ε \end{matrix} ]

as claimed. Similar computations show that for the groups E = E_{E_{6}}^{2b}, E_{E_{6}}^{3a}, E_{E_{6}}^{3c} and E_{E_{6}}^{3d}, the group W′(E) occurring in the Theorem is a lower bound for the Weyl group W (E).

For the group E_{E_{6}}^{3b} = ⟨\overline{a}, \overline{x_{2}}, \overline{y_{2}}⟩ we know the structure of W (π^{−1}(E_{E_{6}}^{3b})) = W (E_{3E_{6}}^{4}) by Theorem 8.7. From this we immediately get W (E_{E_{6}}^{3b}) = SL_{3}(F_{3}).

Exact Weyl groups: We now prove that the lower bounds on the Weyl groups established above are in fact equalities. By Lemma 8.8 the Weyl groups W (E_{E_{6}}^{2a}) and W (E_{E_{6}}^{2b}) are subgroups of SL_{2}(F_{3}). From this we see that W (E_{E_{6}}^{2b}) = SL_{2}(F_{3}) and that W (E_{E_{6}}^{2a}) is equal to either W′(E_{E_{6}}^{2a}) or SL_{2}(F_{3}), since these are the only subgroups of SL_{2}(F_{3}) containing W′(E_{E_{6}}^{2a}). We have E_{E_{6}}^{2a} = ⟨\overline{y_{1}}, \overline{x_{2}}⟩, and by 8.6 we see that the elements \overline{y_{1}} and \overline{x_{2}} are not conjugate in E_{6}(C). In particular we see that W (E_{E_{6}}^{2a}) cannot act transitively on the non-trivial elements of E_{E_{6}}^{2a}, and we conclude that W (E_{E_{6}}^{2a}) = W′(E_{E_{6}}^{2a}) is the group from above.

For each of the remaining non-toral subgroups we now show that a strictly larger Weyl group contradicts the Weyl group results already established. The groups E = E_{E_{6}}^{3a}, E_{E_{6}}^{3d}, and E_{E_{6}}^{4b} all contain E_{E_{6}}^{2a}. A direct computation shows that any proper overgroup of W′(E) in GL(E) contains an element which normalizes the subgroup E_{E_{6}}^{2a} and induces an automorphism which does not lie in W (E_{E_{6}}^{2a}). Hence W (E) = W′(E). If E = E_{E_{6}}^{3c} a similar argument, using the subgroup E_{E_{6}}^{2b}, again shows that W (E) = W′(E). Consider finally E = E_{E_{6}}^{4a}. Each proper overgroup of W′(E) contains an element which normalizes one of the subgroups E_{E_{6}}^{2a} or E_{E_{6}}^{3b} and induces an automorphism on it not contained in its Weyl group. Hence W (E) = W′(E). This concludes the proof that the Weyl groups listed in the theorem are the correct ones.

Centralizers: Let Θ : SL_{3}(C) ⟶ SL_{3}(C)^{3}/C_{3} ⊆ 3E_{6}(C) denote the homomorphism given by Θ(g) = [g, g, g] for g ∈ SL_{3}(C). By Lemma 8.5 the group E_{E_{6}}^{2a} = ⟨\overline{x_{2}}, \overline{y_{1}}⟩ is conjugate to the group ⟨\overline{a}, \overline{s_{1}}⟩. Since a^{s_{1}} = az^{2} we obtain C_{E_{6}(C)}(\overline{a}) = \overline{⟨s_{1}, SL_{3}(C)^{3}/C_{3}⟩}, and hence

C_{E_{6}(C)}(\overline{a}, \overline{s_{1}}) = \overline{⟨a, s_{1}, z, Θ(SL_{3}(C))⟩} = \overline{⟨a, s_{1}, z⟩ × Θ(SL_{3}(C))} = ⟨\overline{a}, \overline{s_{1}}⟩ × PSL_{3}(C),

proving the claims for E_{E_{6}}^{2a}. By abusing the notation slightly, we let \overline{g} denote the image of g ∈ SL_{3}(C) in the quotient PSL_{3}(C). From Lemma 8.5 we then see that the elements \overline{a}, \overline{y_{2}} and \overline{x_{2}x_{1}^{−1}} in C_{E_{6}(C)}(E_{E_{6}}^{2a}) correspond to the elements \overline{β}, \overline{γ^{2}} and \overline{τ_{2}} in the PSL_{3}(C) component of C_{E_{6}(C)}(E_{E_{6}}^{2a}). Thus we immediately get

C_{E_{6}(C)}(E_{E_{6}}^{3a}) = E_{E_{6}}^{2a} × C_{PSL_{3}(C)}(\overline{β}),  C_{E_{6}(C)}(E_{E_{6}}^{3d}) = E_{E_{6}}^{2a} × C_{PSL_{3}(C)}(\overline{τ_{2}}),
C_{E_{6}(C)}(E_{E_{6}}^{4a}) = E_{E_{6}}^{2a} × C_{PSL_{3}(C)}(\overline{β}, \overline{γ^{2}}),  C_{E_{6}(C)}(E_{E_{6}}^{4b}) = E_{E_{6}}^{2a} × C_{PSL_{3}(C)}(\overline{β}, \overline{τ_{2}}).

Note that C_{PSL_{3}(C)}(\overline{β}) = \overline{T_{2} : ⟨γ⟩}, giving C_{PSL_{3}(C)}(\overline{β}, \overline{γ^{2}}) = ⟨\overline{β}, \overline{γ}⟩ and C_{PSL_{3}(C)}(\overline{β}, \overline{τ_{2}}) = \overline{T_{2}}. From this the results on E_{E_{6}}^{3a}, E_{E_{6}}^{4a} and E_{E_{6}}^{4b} follow directly. Note also that C_{PSL_{3}(C)}(\overline{τ_{2}}) ≅ GL_{2}(C) from which we deduce the claims about E_{E_{6}}^{3d}.

Now consider the group E_{E_{6}}^{3b}. Since C_{E_{6}(C)}(\overline{a}) = \overline{⟨s_{1}, SL_{3}(C)^{3}/C_{3}⟩} we get

C_{E_{6}(C)}(\overline{a}, \overline{x_{2}}) = \overline{⟨s_{1}, y_{1}, y_{2}, (T_{2} × T_{2} × T_{2}) /C_{3}⟩},
C_{E_{6}(C)}(\overline{a}, \overline{x_{2}}, \overline{y_{2}}) = \overline{⟨s_{1}, y_{1}, y_{2}, [I_{3}, β, β^{2}] , x_{2}, (⟨ωI_{3}⟩ × ⟨ωI_{3}⟩ × ⟨ωI_{3}⟩) /C_{3}⟩}

and thus C_{E_{6}(C)}(E_{E_{6}}^{3b}) = ⟨E_{E_{6}}^{3b}, \overline{s_{1}}, \overline{y_{1}}, \overline{[I_{3}, β, β^{2}]}⟩. It is now easy to check that C_{E_{6}(C)}(E_{E_{6}}^{3b})

has the structure E_{E_{6}}^{3b} · 3^{3} and that Z(C_{E_{6}(C)}(E_{E_{6}}^{3b})) = E_{E_{6}}^{3b}. For the group E_{E_{6}}^{3c} we obtain

C_{E_{6}(C)}(\overline{a}, \overline{x_{1}}) = \overline{⟨y_{1}, (SL_{3}(C) × T_{2} × T_{2}) /C_{3}⟩},
C_{E_{6}(C)}(\overline{a}, \overline{x_{1}}, \overline{y_{1}}) = \overline{⟨y_{1}, x_{1}, (SL_{3}(C) × ⟨ωI_{3}⟩ × ⟨ωI_{3}⟩) /C_{3}⟩}.

Thus the centralizer C_{E_{6}(C)}(E_{E_{6}}^{3c}) equals the central product E_{E_{6}}^{3c} ∘_{⟨\overline{a}⟩} SL_{3}(C) and we obtain the claims about E_{E_{6}}^{3c}.

Finally we consider the group E_{E_{6}}^{2b} = ⟨\overline{y_{1}}, \overline{x_{1}}⟩. If g ∈ π^{−1}(C_{E_{6}(C)}(E_{E_{6}}^{2b})) then [g, y_{1}] , [g, x_{2}] ∈ Z, and since [y_{1}, x_{2}] = z it follows that g ∈ π^{−1}(E_{E_{6}}^{2b}) ∘_{Z} C_{3E_{6}(C)}(π^{−1}(E_{E_{6}}^{2b})). Thus we have C_{E_{6}(C)}(E_{E_{6}}^{2b}) = E_{E_{6}}^{2b} × \overline{C_{3E_{6}(C)}(π^{−1}(E_{E_{6}}^{2b}))}.

A direct computation shows that C_{3E_{6}(C)}(x_{1}) has type T_{2}D_{4} and a basis for the root system of the centralizer is given by {α_{1} + α_{3} + α_{4}, α_{2}, α_{4} + α_{5} + α_{6}, α_{3} + α_{4} + α_{5}}. From this we see that the 2-dimensional torus consists of the elements h(α, 1, γ, 1, α, γ) where α, γ ∈ C^{×}. Moreover we see that C_{3E_{6}(C)}(x_{1}) = T_{2} ∘_{C} Spin(8, C), where the central product is over the group C = Z(Spin(8, C)) = C_{2} × C_{2} which consist of the elements h(α, 1, γ, 1, α, γ), α, γ = ±1.

Let σ denote the automorphism of C_{3E_{6}(C)}(x_{1}) given by conjugation with y_{1}. A direct check shows that the map from C to C given by x ↦ x^{−1}x^{σ} is surjective. It then follows that

C_{3E_{6}(C)}(π^{−1}(E_{E_{6}}^{2b})) = (T_{2} ∘_{C} Spin(8, C))^{σ} = T_{2}^{σ} ∘_{C^{σ}} Spin(8, C)^{σ}.

We have T_{2}^{σ} = ⟨z⟩, so C_{3E_{6}(C)}(π^{−1}(E_{E_{6}}^{2b})) = ⟨z⟩ × Spin(8, C)^{σ}. Using the class distribution of π^{−1}(E_{E_{6}}^{2b}) found above together with [31, Table 2] and Theorem 8.2(6) we find

dim C_{3E_{6}(C)}(π^{−1}(E_{E_{6}}^{2b})) = \frac{1}{3^{3}} · (3 · 78 + 24 · (30 + 24ω + 24ω^{2})) = 14.

Thus Spin(8, C)^{σ} has dimension 14 and since Z(Spin(8, C))^{σ} = 1 we also see that Spin(8, C)^{σ} has rank less than 4. From this it follows that the identity component of Spin(8, C)^{σ} must have type G_{2}. By [120, Thm. 8.1] we know that Spin(8, C)^{σ} is connected, so we get Spin(8, C)^{σ} = G_{2}(C) and hence C_{3E_{6}(C)}(π^{−1}(E_{E_{6}}^{2b})) = ⟨z⟩ × G_{2}(C). Combining this with the computation from above we conclude C_{E_{6}(C)}(E_{E_{6}}^{2b}) = E_{E_{6}}^{2b} × G_{2}(C). □

For the proof of our main results we need the following auxiliary results about the two non-toral elementary abelian 3-groups of rank 2.

Proposition 8.10. Let E be an elementary abelian 3-group of rank 2 with basis (e_{1}, e_{2}) and consider the 4 homomorphisms μ_{i} : E → N_{E_{6}(C)}(\overline{H}) (the maximal torus normalizer in E_{6}(C)), 1 ≤ i ≤ 4 defined as follows:

μ_{1} : e_{1} ↦ \overline{y_{1}}, e_{2} ↦ \overline{x_{2}},
μ_{2} : e_{1} ↦ \overline{y_{1}}, e_{2} ↦ \overline{x_{2}y_{1}},
μ_{3} : e_{1} ↦ \overline{y_{1}}, e_{2} ↦ \overline{x_{2}y_{1}^{−1}},
μ_{4} : e_{1} ↦ \overline{x_{1}}, e_{2} ↦ \overline{x_{2}y_{1}^{−1}}.

Then μ_{i}(E) is conjugate to E_{E_{6}}^{2a} for all i and μ_{i}^{−1}(\overline{H}) equals the 4 distinct rank 1 subgroups of E for i = 1, …, 4. Moreover T_{E} = (\overline{H}^{μ_{i}(E)})_{1} is independent of i and C_{N_{E_{6}(C)}(\overline{H})}(μ_{i}(E)) is a maximal torus normalizer in C_{E_{6}(C)}(μ_{i}(E)) for all i. The canonical homomorphism E × T_{E} ⟶^{μ_{i}×1} N(H) → E_{6}(C) does not depend on i up to conjugacy in E_{6}(C).

Proof. Obviously μ_{1}^{−1}(\overline{H}) = ⟨e_{2}⟩, μ_{2}^{−1}(\overline{H}) = ⟨e_{1} − e_{2}⟩, μ_{3}^{−1}(\overline{H}) = ⟨e_{1} + e_{2}⟩ and μ_{4}^{−1}(\overline{H}) = ⟨e_{1}⟩, so μ_{i}^{−1}(\overline{H}) equals the 4 distinct rank 1 subgroups of E for i = 1, …, 4.

Since \overline{x_{1}}, \overline{x_{2}} ∈ \overline{H}, it is clear that the identity component of \overline{H}^{μ_{i}(E)} equals the identity component of \overline{H}\overline{y_{1}}. Hence T_{E} = (\overline{H}^{μ_{i}(E)})_{1} is independent of i and as y_{1} = [I_{3}, γ, γ^{2}], a direct computation shows that T_{E} consist of the elements of the form \overline{[g, I_{3}, I_{3}]}, g ∈ T_{2}, where T_{2} ⊆ SL_{3}(C) denotes the maximal torus consisting of diagonal matrices.

We now prove that the homomorphisms E ⟶^{μ_{i}} C_{E_{6}(C)}(T_{E}) are conjugate in C_{E_{6}(C)}(T_{E}). As \overline{a} ∈ T_{E} we find that C_{E_{6}(C)}(T_{E}) consists of the elements \overline{[g_{1}, g_{2}, g_{3}]}, where g_{1} ∈ T_{2} and g_{2}, g_{3} ∈ SL_{3}(C) are arbitrary. Note first that the conjugation by the element \overline{[I_{3}, τ_{1}, τ_{1}^{2}]} ∈ C_{E_{6}(C)}(T_{E}) sends \overline{y_{1}} to itself and \overline{x_{2}} to \overline{x_{2}y_{1}}. Hence the homomorphism E ⟶^{μ_{i}} C_{E_{6}(C)}(T_{E}) is conjugate to the homomorphism E ⟶^{μ_{i+1}} C_{E_{6}(C)}(T_{E}) for i = 1, 2. Letting

τ_{3} = −\frac{e^{πi/18}}{\sqrt{3}} [ \begin{matrix}1 & ω^{2} & 1 \\ 1 & 1 & ω^{2} \\ 1 & ω & ω \end{matrix} ] ∈ SL_{3}(C),

we get β^{τ_{3}} = βγ and γ^{τ_{3}} = β^{2}. Thus the conjugation by the element \overline{[I_{3}, τ_{2}τ_{1}^{−1}, τ_{3}]} ∈ C_{E_{6}(C)}(T_{E}) sends \overline{y_{1}} to \overline{x_{1}} and \overline{x_{2}} to \overline{x_{2}y_{1}^{−1}} and hence the homomorphism E ⟶^{μ_{1}} C_{E_{6}(C)}(T_{E}) is conjugate to the homomorphism E ⟶^{μ_{4}} C_{E_{6}(C)}(T_{E}). This proves the claim. We conclude that E × T_{E} ⟶^{μ_{i}×1} N(H) → E_{6}(C) is independent of i up to conjugacy in E_{6}(C) and also that μ_{i}(E) is conjugate to μ_{1}(E) = E_{E_{6}}^{2a} for all i.

Since C_{E_{6}(C)}(E_{E_{6}}^{2a}) has rank 2 it follows that T_{E} is a maximal torus in C_{E_{6}(C)}(μ_{i}(E)) for all i. Since μ_{i}(E) is elementary abelian of rank 2 and μ_{i}(E) ∩ \overline{H} ≠ 1 it now follows from Theorem 8.3 that C_{N_{E_{6}(C)}(\overline{H})}(μ_{i}(E)) is a maximal torus normalizer in C_{E_{6}(C)}(μ_{i}(E)) for all i. □

Proposition 8.11. An element α ∈ W (E_{E_{6}}^{2b}) acts up to conjugacy by α×1 on C_{E_{6}(C)}(E_{E_{6}}^{2b}) = E_{E_{6}}^{2b} × G_{2}(C).

Proof. Since G_{2}(C) is connected and has trivial center it is clear that Out(E_{E_{6}}^{2b} × G_{2}(C)) = Out(E_{E_{6}}^{2b})×Out(G_{2}(C)). The result now follows as Out(G_{2}(C)) = 1 (i.e. by [70, Thm. 27.4]).

□ 8.3. The group E_{8}(C), p = 3. In this section we consider the elementary abelian 3subgroups of the group E_{8}(C). By using [14, Table 2, p. 214] we see that the smallest faithful representation of E_{8}(C) is the adjoint representation, i.e. the representation given by the action of E_{8}(C) on its Lie algebra 𝔢_{8}, which has dimension 248. For our computations, we explicitly construct this representation on a computer by following the recipe in [22, Ch. 4]. (As explained in [22, Ch. 4] there is some ambiguity in choosing a Chevalley basis of 𝔢_{8} and we fix a certain such choice; a different choice affects our formulas at only one point—see Remark 8.12.)

Letting Φ(E_{8}) denote the root system of type E_{8} (we use the notation of [13, p. 268– 270]), we have in particular a maximal torus H generated by the elements h_{α_{i}}(t), 1 ≤ i ≤ 8, t ∈ C^{×} ([22, p. 92, p. 97]) and root subgroups u_{α}(t), α ∈ Φ(E_{8}), t ∈ C. The normalizer N(H) of the maximal torus, is generated by H and the elements n_{i} = n_{α_{i}}, 1 ≤ i ≤ 8 ([22, p. 93, p. 101]). We let

h(t_{1}, t_{2}, t_{3}, t_{4}, t_{5}, t_{6}, t_{7}, t_{8}) = ∏_{i=1}^{8} h_{α_{i}}(t_{i}).

Note that by [22, p. 100 and Lem. 6.4.4] the root subgroups u_{α} form a realization ([118, p. 133]) of Φ(E_{8}) in E_{8}(C). In particular we have the following relations. The element n_{i} has image s_{α_{i}} in W = W (E_{8}) ([118, 8.1.4(i)]), we have n_{i}^{2} = h_{α_{i}}(−1) ([118, 8.1.4(ii)]) and

n_{i}n_{j}n_{i} … = n_{j}n_{i}n_{j} …

for 1 ≤ i, j ≤ 8, where the number of factors on both sides equals the order of s_{α_{i}}s_{α_{j}} in W ([118, 9.3.2]).

Now let \overline{a} = h_{α_{1}}(ω)h_{α_{2}}(ω)h_{α_{3}}(ω^{2}) ∈ E_{8}(C). Direct computation shows that for any root α ∈ Φ(E_{8}) we have α(\overline{a}) = ω^{2⟨α,λ_{2}⟩}. From this we see that Dynkin diagram of the centralizer C_{E_{8}(C)}(\overline{a}) is the same as the extended Dynkin diagram of E_{8} with the node α_{2} removed. Thus it has type A_{8} and basis

{α_{1}, α_{3}, α_{4}, α_{5}, α_{6}, α_{7}, α_{8}, −α̃},

where α̃ is the longest root. As in [13, p. 250–251] we identify Φ(A_{8}) with the set of elements in R^{9} of the form e_{i}−e_{j} with i ≠ j and 1 ≤ i, j ≤ 9, where e_{i} denotes the i’th canonical basis vector in R^{9}. We now consider SL_{9}(C), which is the simply connected group of type A_{8} over C. For any 1 ≤ i, j ≤ 9 we let e_{i,j} be the 9 × 9-matrix over C whose only non-zero element is 1 occurring at the (i, j)-entry. Given a root α′ = e_{i} − e_{j} ∈ Φ(A_{8}) we let u′_{α′}(t) = I_{9} + te_{i,j} for t ∈ C. With respect to the maximal torus consisting of the diagonal matrices, this is a root subgroup of SL_{9}(C) corresponding to the root α′. The roots α′_{i} = e_{i} − e_{i+1}, 1 ≤ i ≤ 8, form a basis of Φ(A_{8}). From the above we then see that

u′_{±α′_{1}}(t) ↦ u_{±α_{1}}(t),  u′_{±α′_{2}}(t) ↦ u_{±α_{3}}(t),  u′_{±α′_{3}}(t) ↦ u_{±α_{4}}(t),  u′_{±α′_{4}}(t) ↦ u_{±α_{5}}(t),
u′_{±α′_{5}}(t) ↦ u_{±α_{6}}(t),  u′_{±α′_{6}}(t) ↦ u_{±α_{7}}(t),  u′_{±α′_{7}}(t) ↦ u_{±α_{8}}(t),  u′_{±α′_{8}}(t) ↦ u_{∓α̃}(t)

defines a homomorphism SL_{9}(C) → E_{8}(C) onto the centralizer C_{E_{8}(C)}(\overline{a}). It is easy to check that this map has kernel C_{3} = ⟨ωI_{9}⟩ and thus we may make the identification C_{E_{8}(C)}(\overline{a}) = SL_{9}(C)/C_{3}. For any g ∈ SL_{9}(C) we denote by \overline{g} its image in SL_{9}(C)/C_{3} = C_{E_{8}(C)}(\overline{a}) ⊆ E_{8}(C). In particular we see that a = ηI_{9} corresponds to the element \overline{a} from above. We also define the following elements in SL_{9}(C) :

x_{1} = diag(1, ω, ω^{2}, 1, ω, ω^{2}, 1, ω, ω^{2}),  x_{2} = diag(1, 1, 1, ω, ω, ω, ω^{2}, ω^{2}, ω^{2}),
x_{3} = diag(1, 1, 1, 1, 1, 1, ω, ω, ω),  y_{1} = (1, 2, 3)(4, 5, 6)(7, 8, 9),
y_{2} = (1, 4, 7)(2, 5, 8)(3, 6, 9).

From the explicit homomorphism above we easily find

\overline{a} = h_{α_{1}}(ω)h_{α_{2}}(ω)h_{α_{3}}(ω^{2}),  \overline{x_{1}} = h_{α_{1}}(ω)h_{α_{5}}(ω)h_{α_{8}}(ω),
\overline{x_{2}} = h_{α_{1}}(ω)h_{α_{3}}(ω^{2})h_{α_{5}}(ω^{2})h_{α_{6}}(ω),  \overline{x_{3}} = h_{α_{1}}(ω^{2})h_{α_{3}}(ω)h_{α_{5}}(ω^{2})h_{α_{6}}(ω),

and a direct computation in E_{8}(C) shows that

\begin{aligned}n_{−α̃} = & n_{8}n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{2}n_{6}n_{5}n_{4}n_{3}n_{1}n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{4}n_{5}n_{6}n_{7}n_{8}· \\ & n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{2}n_{6}n_{5}n_{4}n_{3}n_{1}n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{4}n_{5}n_{6}n_{7}n_{8}. \end{aligned}

Remark 8.12. A different choice of Chevalley basis for 𝔢_{8} may effect the expression for n_{−α̃} by a order two element in H. If a Chevalley basis is chosen such that the above formula holds then all further formulas will be independent of the choice.

From this and the explicit homomorphism above we find, either by direct computation or by using the relations in N(H), that

y_{1} = n_{1}n_{3}n_{5}n_{6}n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{2}n_{6}n_{5}n_{4}n_{3}n_{1}n_{7}n_{6}n_{5}·
n_{4}n_{2}n_{3}n_{4}n_{5}n_{6}n_{7}n_{8}n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{2}n_{6}n_{5}n_{4}n_{3}·
n_{1}n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{4}n_{5}n_{6}n_{7}n_{8} · h_{α_{1}}(−1)h_{α_{2}}(−1)h_{α_{7}}(−1),
y_{2} = n_{2}n_{3}n_{1}n_{4}n_{2}n_{3}n_{4}n_{5}n_{4}n_{2}n_{3}n_{4}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{7}n_{6}n_{5}n_{4}·
n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{6}n_{7}n_{8}n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{2}n_{6}n_{5}·
n_{4}n_{3}n_{1}n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{4}n_{5}n_{8}n_{7}n_{6} · h_{α_{2}}(−1)h_{α_{5}}(−1).

Notation 8.13. To distinguish subgroups of E_{8}(C), we need some information on the conjugacy classes of elements of order 3. These are given in [63, Table VI] (which is taken from [30, Table 4]): There are 4 such conjugacy classes, which we label 3A, 3B, 3C and 3D. Moreover these classes may be distinguished by their traces on 𝔢_{8}. Since the trace of the element h ∈ H is given by 8 + ∑_{α∈Φ(E_{8})} α(h) we get \overline{a} ∈ 3A, \overline{x_{1}}, \overline{x_{2}}, \overline{x_{3}}, \overline{y_{1}}, \overline{y_{2}} ∈ 3B and \overline{x_{3}a^{−1}} ∈ 3D.

Notation 8.14. If K is a field and n is a natural number, we define the group of symplectic similitudes as CSp_{2n}(K) = {X ∈ GL_{2n}(K)|X^{t}BX = cB, c ∈ K^{×}}, where

B = \underbrace{[ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ] ⊕ … ⊕ [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ]}_{n times}

We define the homomorphism χ : CSp_{2n}(K) → K^{×} by χ(X) = c, where X^{t}BX = cB. The kernel of χ is the symplectic group Sp_{2n}(K). (The notation CSp is taken from [82].)

Theorem 8.15. The conjugacy classes of non-toral elementary abelian 3-subgroups of E_{8}(C) are given by the following table.

rank name

ordered basis  E_{8}(C)-class dist.  C_{E_{8}(C)}(E)  Z(C_{E_{8}(C)}(E))
3 E_{E_{8}}^{3a}  ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{a}⟩  3A^{18}3B^{8}  E_{E_{8}}^{3a} × PSL_{3}(C)  E_{E_{8}}^{3a}
3 E_{E_{8}}^{3b}  ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{x_{3}}⟩  3B^{26}  E_{E_{8}}^{3b} × G_{2}(C)  E_{E_{8}}^{3b}
4 E_{E_{8}}^{4a}  ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{x_{3}}, \overline{x_{3}a^{−1}}⟩  3A^{52}3B^{26}3D^{2} E_{E_{8}}^{4a} ∘_{⟨\overline{x_{3}a^{−1}}⟩} GL_{2}(C) E_{E_{8}}^{4a} ∘_{⟨\overline{x_{3}a^{−1}}⟩} T_{1}
4 E_{E_{8}}^{4b}  ⟨\overline{x_{2}}, \overline{x_{1}}, \overline{y_{1}}, \overline{a}⟩  3A^{54}3B^{26}  E_{E_{8}}^{4b} ∘_{⟨\overline{x_{2}}⟩} (T_{2} : ⟨\overline{y_{2}}⟩)  E_{E_{8}}^{4b}
4 E_{E_{8}}^{4c}  ⟨\overline{x_{2}}, \overline{x_{1}}, \overline{y_{1}}, \overline{x_{3}}⟩  3B^{80}  E_{E_{8}}^{4c} ∘_{⟨\overline{x_{2}}⟩} SL_{3}(C)  E_{E_{8}}^{4c}
5 E_{E_{8}}^{5a} ⟨\overline{x_{2}}, \overline{x_{1}}, \overline{y_{1}}, \overline{x_{3}}, \overline{x_{3}a^{−1}}⟩ 3A^{156}3B^{80}3D^{6} E_{E_{8}}^{5a} ∘_{⟨\overline{x_{2}},\overline{x_{3}a^{−1}}⟩} T_{2} E_{E_{8}}^{5a} ∘_{⟨\overline{x_{2}},\overline{x_{3}a^{−1}}⟩} T_{2}
5 E_{E_{8}}^{5b}  ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{x_{2}}, \overline{y_{2}}, \overline{a}⟩  3A^{162}3B^{80}  E_{E_{8}}^{5b}  E_{E_{8}}^{5b}

In particular we have _{3}Z(C_{E_{8}(C)}(E)) = E for any non-toral elementary abelian 3-subgroup of E_{8}(C).

The Weyl groups of these groups with respect to the given ordered bases are given as follows:

W (E_{E_{8}}^{3a}) = [ \begin{matrix}GL_{2}(F_{3}) & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 0 & det \end{matrix} ] , W (E_{E_{8}}^{3b}) = SL_{3}(F_{3}), W (E_{E_{8}}^{4a}) = [ \begin{matrix}SL_{3}(F_{3}) & 0 0 0 \\ 0 0 0 & ε \end{matrix} ]

W (E_{E_{8}}^{4b}) = [ \begin{matrix}ε & ∗ ∗ & ∗ \\ {0 \atop 0} & GL_{2}(F_{3}) & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 & 0 0 & det \end{matrix} ] , W (E_{E_{8}}^{4c}) = [ \begin{matrix}ε & ∗ ∗ ∗ \\ 0 0 0 & SL_{3}(F_{3}) \end{matrix} ]

W (E_{E_{8}}^{5a}) = [ \begin{matrix}ε_{1} & ∗ ∗ ∗ & ∗ \\ 0 0 0 & SL_{3}(F_{3}) & 0 0 0 \\ 0 & 0 0 0 & ε_{2} \end{matrix} ] , W (E_{E_{8}}^{5b}) = [ \begin{matrix}CSp_{4}(F_{3}) & \begin{matrix}∗ \\ ∗ \\ ∗ \\ ∗ \end{matrix} \\ 0 0 0 0 & χ \end{matrix} ]

where det is the determinant of the matrix from GL_{2}(F_{3}) in the description of W (E_{E_{8}}^{3a}) and W (E_{E_{8}}^{4b}). In the description of W (E_{E_{8}}^{5b}), χ denotes the value of the homomorphism χ : CSp_{4}(F_{3}) → F_{3}^{×} defined in 8.14 evaluated on the matrix from CSp_{4}(F_{3}).

Remark 8.16. Note that our information on the rank five subgroup E_{E_{8}}^{5a} corrects [63].

Proof of Theorem 8.15. Maximal non-toral subgroups: By [63, Lems. 11.7 and 11.9], any maximal non-toral elementary abelian 3-subgroup of E_{8}(C) contains an element of type 3A. We may thus find representatives in C_{E_{8}(C)}(\overline{a}) = SL_{9}(C)/C_{3}. From [63, Cor. 11.10], it follows that there are two conjugacy classes of these maximal non-toral elementary abelian 3-subgroups: E_{E_{8}}^{5a} and E_{E_{8}}^{5b}, both of which have rank 5. Moreover, by [63, Lem. 11.5], their preimages in SL_{9}(C) may be chosen to have the shape 3^{1+2} ∘_{C_{3}} 9 × 3 × 3 and 3^{1+4} ∘_{C_{3}} 9. Using the representation theory of extraspecial 3-groups ([62, Ch. 5.5]) we find that E_{E_{8}}^{5a} is

represented by ⟨\overline{x_{2}}, \overline{x_{1}}, \overline{y_{1}}, \overline{x_{3}}, \overline{x_{3}a^{−1}}⟩ and that E_{E_{8}}^{5b} is represented by ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{x_{2}}, \overline{y_{2}}, \overline{a}⟩.

Lower bounds for Weyl groups of maximal non-toral subgroups: We can find lower bounds for the Weyl groups of E_{E_{8}}^{5a} and E_{E_{8}}^{5b} by conjugating with elements in the centralizer C_{E_{8}(C)}(\overline{a}) = SL_{9}(C)/C_{3} and the normalizer N(H) of the maximal torus.

Note that

a = ηI_{3} ⊕ ηI_{3} ⊕ ηI_{3},  x_{1} = β ⊕ β ⊕ β,  x_{2} = I_{3} ⊕ ωI_{3} ⊕ ω^{2}I_{3},
x_{3} = I_{3} ⊕ I_{3} ⊕ ωI_{3},  y_{1} = γ ⊕ γ ⊕ γ

and (A ⊕ B ⊕ C)^{y_{2}} = B ⊕ C ⊕ A. Conjugation by τ_{1} ⊕ τ_{1} ⊕ τ_{1}, τ_{2} ⊕ τ_{2} ⊕ τ_{2} and I_{3} ⊕ β^{2} ⊕ β gives

(8.1) τ_{1} ⊕ τ_{1} ⊕ τ_{1} : a ↦ a, x_{1} ↦ x_{1}y_{1}, x_{2} ↦ x_{2}, x_{3} ↦ x_{3}, y_{1} ↦ y_{1}, y_{2} ↦ y_{2}.
(8.2) τ_{2} ⊕ τ_{2} ⊕ τ_{2} : a ↦ a, x_{1} ↦ x_{1}, x_{2} ↦ x_{2}, x_{3} ↦ x_{3}, y_{1} ↦ x_{1}y_{1}, y_{2} ↦ y_{2}.
(8.3) I_{3} ⊕ β^{2} ⊕ β : a ↦ a, x_{1} ↦ x_{1}, x_{2} ↦ x_{2}, x_{3} ↦ x_{3}, y_{1} ↦ x_{2}y_{1}, y_{2} ↦ x_{1}y_{2}.

Now consider the group E_{E_{8}}^{5a}. From (8.1)—(8.3) we see that the elements \overline{τ_{1} ⊕ τ_{1} ⊕ τ_{1}}, \overline{τ_{2} ⊕ τ_{2} ⊕ τ_{2}} and \overline{I_{3} ⊕ β^{2} ⊕ β} normalize E_{E_{8}}^{5a} and that conjugation by these elements induces the automorphisms on E_{E_{8}}^{5a} given by the matrices I_{5} + e_{3,2}, I_{5} + e_{2,3} and I_{5} + e_{1,3}.

Letting σ = −(1, 4)(2, 5)(3, 6) ∈ SL_{9}(C) we see that (A ⊕ B ⊕ C)^{σ} = B ⊕ A ⊕ C. Using this and the above we obtain that \overline{σ}, \overline{y_{2}} and \overline{I_{3} ⊕ I_{3} ⊕ β^{2}} normalize E_{E_{8}}^{5a} and that conjugation by these elements induces the automorphisms on E_{E_{8}}^{5a} given by the matrices diag(2, 1, 1, 1, 1), I_{5} + e_{1,4} + e_{1,5} and I_{5} + e_{4,3}. By using the relations in N(H) given above or by direct computation, it may be checked that conjugation by the element

\begin{aligned}n = & n_{1}n_{2}n_{4}n_{2}n_{3}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{6}n_{5}n_{4}n_{2}n_{3}n_{4}n_{7}n_{6}n_{5}n_{4}n_{8}n_{7}n_{6}· \\ & n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{2}n_{6}n_{5}n_{4}n_{7} · h(1, 1, −1, −1, −1, 1, −1, −1) \end{aligned}

induces the automorphism on E_{E_{8}}^{5a} represented by the matrix diag(1, 1, 1, 1, 2) + e_{2,4}. It is easy to see that the above matrices generate the group

W′(E_{E_{8}}^{5a}) = [ \begin{matrix}ε_{1} & ∗ ∗ ∗ & ∗ \\ 0 0 0 & SL_{3}(F_{3}) & 0 0 0 \\ 0 & 0 0 0 & ε_{2} \end{matrix} ]

and thus W (E_{E_{8}}^{5a}) contains this group.

Next consider the group E_{E_{8}}^{5b}. From (8.1)—(8.3) we see that the elements \overline{τ_{1} ⊕ τ_{1} ⊕ τ_{1}}, \overline{τ_{2} ⊕ τ_{2} ⊕ τ_{2}} and \overline{I_{3} ⊕ β^{2} ⊕ β} normalize E_{E_{8}}^{5b} and that conjugation by these elements induces the automorphisms on E_{E_{8}}^{5b} given by the matrices I_{5} + e_{2,1}, I_{5} + e_{1,2} and I_{5} + e_{1,4} + e_{3,2}. Now note that a = Δ_{3,3}(ηI_{3}), x_{2} = Δ_{3,3}(β) and y_{2} = Δ_{3,3}(γ). Noting also that Δ_{3,3}(M_{1}) commutes with M_{2} ⊕ M_{2} ⊕ M_{2} for any M_{1}, M_{2} ∈ M_{3}(C) we see that the elements \overline{Δ_{3,3}(τ_{1})} and \overline{Δ_{3,3}(τ_{2})} normalize E_{E_{8}}^{5b}. The automorphisms induced on E_{E_{8}}^{5b} by conjugation with these elements have the matrices I_{5} + e_{4,3} and I_{5} + e_{3,4}. The upper left 4 × 4-corner of these matrices are easily seen to generate the group Sp_{4}(F_{3}) from 8.14. By using the relations in N(H) given above or by direct computation, we get that conjugation by the element

\begin{aligned}n = & n_{2}n_{8}n_{7}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{2}n_{6}n_{5}n_{4}n_{3}n_{1}n_{7}n_{6}· \\ & n_{5}n_{4}n_{2}n_{3}n_{4}n_{5}n_{6}n_{7}n_{8} · h(1, −1, −1, −1, −1, 1, 1, 1) \end{aligned}

of N(H) induces the automorphism on E_{E_{8}}^{5b} represented by the matrix diag(1, 2, 1, 2, 2)+e_{3,5}. It now follows that W (E_{E_{8}}^{5b}) contains the group

W′(E_{E_{8}}^{5b}) = [ \begin{matrix}CSp_{4}(F_{3}) & \begin{matrix}∗ \\ ∗ \\ ∗ \\ ∗ \end{matrix} \\ 0 0 0 0 & χ \end{matrix} ]

Lower bounds for other Weyl groups: We now show that the other Weyl groups in the table are all lower bounds. To do this consider one of the non-maximal groups E from the table. We then have E ⊆ E_{E_{8}}^{5a}, and we get a lower bound on W (E) by considering the action on E of the subgroup of W′(E_{E_{8}}^{5a}) fixing E. As an example we find that the stabilizer of E_{E_{8}}^{3a} inside W′(E_{E_{8}}^{5a}) is

[ \begin{matrix}ε_{1} & 0 & 0 & x & x \\ 0 & a_{11} & a_{12} & a_{13} & 0 \\ 0 & a_{21} & a_{22} & a_{23} & 0 \\ 0 & 0 & 0 & det & 0 \\ 0 & 0 & 0 & 0 & det \end{matrix} ]

where det = a_{11} · a_{22} − a_{12} · a_{21} ≠ 0. The action of such a matrix on E_{E_{8}}^{3a} is given by

\overline{x_{1}} ↦ (\overline{x_{1}})^{a_{11}}(\overline{y_{1}})^{a_{21}},  \overline{y_{1}} ↦ (\overline{x_{1}})^{a_{12}}(\overline{y_{1}})^{a_{22}},  \overline{a} ↦ (\overline{x_{1}})^{a_{13}}(\overline{y_{1}})^{a_{23}}(\overline{a})det.

Thus W (E_{E_{8}}^{3a}) contains the group

W′(E_{E_{8}}^{3a}) = [ \begin{matrix}GL_{2}(F_{3}) & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 0 & det \end{matrix} ]

as claimed. Similar computations show that for the remaining groups E = E_{E_{8}}^{3b}, E_{E_{8}}^{4a}, E_{E_{8}}^{4b} and E_{E_{8}}^{4c}, the group W′(E) occurring in the Theorem is a lower bound for the Weyl group W (E).

Orbit computation: Note first that all elementary abelian subgroups of rank at most two are toral by Theorem 8.2(3). By using the lower bounds on the Weyl groups of E_{E_{8}}^{5a} and E_{E_{8}}^{5b} established above, we may find a set of representatives for the conjugacy classes of subgroups of E_{E_{8}}^{5a} and E_{E_{8}}^{5b} of rank 3 and 4.

Under the action of W′(E_{E_{8}}^{5a}), the set of rank 3 subgroups of E_{E_{8}}^{5a} has orbit representatives

\begin{matrix}E_{E_{8}}^{3a} = ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{a}⟩ , E_{E_{8}}^{3b} = ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{x_{3}}⟩ , ⟨\overline{x_{1}}, \overline{x_{2}}, \overline{y_{1}}⟩ , \\ ⟨\overline{a}, \overline{x_{1}}, \overline{x_{2}}⟩ , ⟨\overline{a}, \overline{x_{1}}, \overline{x_{3}}⟩ and ⟨\overline{a}, \overline{x_{2}}, \overline{x_{3}}⟩ , \end{matrix}

and under the action of W′(E_{E_{8}}^{5b}), the set of rank 3 subgroups of E_{E_{8}}^{5b} has orbit representatives

E_{E_{8}}^{3a} = ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{a}⟩ , ⟨\overline{x_{1}}, \overline{x_{2}}, \overline{y_{1}}⟩ and ⟨\overline{a}, \overline{x_{1}}, \overline{x_{2}}⟩ .

Similarly we find that under the action of W′(E_{E_{8}}^{5a}), the set of rank 4 subgroups of E_{E_{8}}^{5a} has orbit representatives

\begin{matrix}E_{E_{8}}^{4a} = ⟨\overline{x_{1}}, \overline{y_{1}}, \overline{x_{3}}, \overline{x_{3}a^{−1}}⟩ , E_{E_{8}}^{4b} = ⟨\overline{x_{2}}, \overline{x_{1}}, \overline{y_{1}}, \overline{a}⟩ , \\ E_{E_{8}}^{4c} = ⟨\overline{x_{2}}, \overline{x_{1}}, \overline{y_{1}}, \overline{x_{3}}⟩ and ⟨\overline{a}, \overline{x_{1}}, \overline{x_{2}}, \overline{x_{3}}⟩ , \end{matrix}

and that under the action of W′(E_{E_{8}}^{5b}), the set of rank 4 subgroups of E_{E_{8}}^{5b} has orbit representatives

E_{E_{8}}^{4b} = ⟨\overline{x_{2}}, \overline{x_{1}}, \overline{y_{1}}, \overline{a}⟩ and E_{0} = ⟨\overline{x_{1}}, \overline{x_{2}}, \overline{y_{1}}, \overline{y_{2}}⟩ .

Class distributions: Recall that by 8.13, \overline{a} is in the conjugacy class 3A, \overline{x_{1}} and \overline{x_{2}} are in the class 3B and \overline{x_{3}a^{−1}} belongs to the class 3D. Using the actions of W′(E_{E_{8}}^{5a}) and W′(E_{E_{8}}^{5b}) it is then straightforward to verify the class distributions given in the table. As an example consider the group E_{E_{8}}^{5a}. Under the action of W′(E_{E_{8}}^{5a}) it contains 156 elements conjugate to \overline{a}, 78 elements conjugate to \overline{x_{1}}, 2 elements conjugate to \overline{x_{2}} and 6 elements conjugate to \overline{x_{3}a^{−1}}, which gives the class distribution in the table. Similar computations give the results for the remaining groups.

We also see that the group E_{0} = ⟨\overline{x_{1}}, \overline{x_{2}}, \overline{y_{1}}, \overline{y_{2}}⟩ has class distribution 3B^{80} and from the class distribution of E_{E_{8}}^{5b} we get E_{0} = (E_{E_{8}}^{5b} ∩ 3B) ∪ {1}. It then follows from [63, Lem. 11.5] that E_{0} is toral.

Other non-toral subgroups: We see directly that the groups

⟨\overline{a}, \overline{x_{1}}, \overline{x_{2}}, \overline{x_{3}}⟩ , ⟨\overline{a}, \overline{x_{1}}, \overline{x_{2}}⟩ , ⟨\overline{a}, \overline{x_{1}}, \overline{x_{3}}⟩ and ⟨\overline{a}, \overline{x_{2}}, \overline{x_{3}}⟩

are toral. Since the group ⟨\overline{x_{1}}, \overline{x_{2}}, \overline{y_{1}}⟩ is a subgroup of E_{0} it is also toral. Alternatively, from the action of W′(E_{E_{8}}^{5a}) we see that it is conjugate to the group ⟨\overline{x_{1}}, \overline{x_{2}}, \overline{x_{3}}⟩, which is visibly toral. Thus any non-toral elementary abelian 3-subgroup of E_{8}(C) is conjugate to a group in the table. Moreover, since their class distributions differ, none of the groups occurring in the table are conjugate.

To see that the groups in the table are actually non-toral we may proceed as follows. The group E_{E_{8}}^{3a} contains the element \overline{a}, so by Theorem 8.2(1) it is toral if and only if it is toral in C_{E_{8}(C)}(\overline{a}) = SL_{9}(C)/C_{3}. However this is not the case by Theorem 8.2(5), since its lift to SL_{9}(C) is non-abelian. The groups E_{E_{8}}^{4a} and E_{E_{8}}^{4b} are thus also non-toral since they contain E_{E_{8}}^{3a}. We saw above that the Weyl group of E_{E_{8}}^{3b} contains SL_{3}(F_{3}), which has order 2^{4} · 3^{3} · 13. Since the Weyl group of E_{8}(C) has order 2^{14} · 3^{5} · 5^{2} · 7 which is not divisible by 13 it follows from Theorem 8.2(2) that E_{E_{8}}^{3b} is non-toral. Since E_{E_{8}}^{4c} contains E_{E_{8}}^{3b} it is also non-toral.

Centralizers: The subgroups E = E_{E_{8}}^{3a}, E_{E_{8}}^{4a}, E_{E_{8}}^{4b}, E_{E_{8}}^{5a} and E_{E_{8}}^{5b} are easy to deal with since they all contain \overline{a}, and hence we have C_{E_{8}(C)}(E) = C_{SL_{9}(C)/C_{3}}(E) for these. It is however notationally convenient first to change the representatives as follows. Define x_{4} = τ_{2}^{−1} ⊕τ_{2}^{−1} ⊕τ_{2}^{−1} ∈ SL_{9}(C), and note that conjugation by (2, 7, 3, 4)(5, 8, 9, 6) ∈ SL_{9}(C) acts as follows:

a ↦ a,  x_{1} ↦ x_{2},  x_{2} ↦ x_{1}^{2},  x_{3}a^{−1} ↦ x_{4},  y_{1} ↦ y_{2},  y_{2} ↦ y_{1}^{2}.

In particular we see that E_{E_{8}}^{3a} is conjugate to ⟨\overline{x_{2}}, \overline{y_{2}}, \overline{a}⟩. Moreover we have

C_{E_{8}(C)}(\overline{a}, \overline{x_{2}}) = \overline{⟨y_{2}, \{A ⊕ B ⊕ C | det ABC = 1\}⟩}.

From this we directly get

C_{E_{8}(C)}(\overline{a}, \overline{x_{2}}, \overline{y_{2}}) = \overline{⟨x_{2}, y_{2}, \{A ⊕ A ⊕ A | (det A)^{3} = 1\}⟩}
= \overline{⟨x_{2}, y_{2}, a, \{A ⊕ A ⊕ A | det A = 1\}⟩}
≅ ⟨\overline{x_{2}}, \overline{y_{2}}, \overline{a}⟩ × PSL_{3}(C).

Thus C_{E_{8}(C)}(E_{E_{8}}^{3a}) = E_{E_{8}}^{3a} × PSL_{3}(C) and Z(C_{E_{8}(C)}(E_{E_{8}}^{3a})) = E_{E_{8}}^{3a}. From the above we see

that the elements \overline{x_{2}}, \overline{x_{3}a^{−1}} and \overline{y_{2}} in C_{E_{8}(C)}(E_{E_{8}}^{3a}) correspond to the elements \overline{β^{2}}, \overline{τ_{2}^{−1}} and \overline{γ^{2}} in the PSL_{3}(C) component of C_{E_{8}(C)}(E_{E_{8}}^{3a}). From this we easily compute the structure of C(E) for the representatives E which contain E_{E_{8}}^{3a}, cf. the proof of Theorem 8.9.

For the computation of the centralizers of E_{E_{8}}^{3b} and E_{E_{8}}^{4c} we consider the element g = h_{α_{1}}(ω)h_{α_{3}}(ω^{2}) ∈ E_{8}(C). By using [30, Tables 4 and 6] we get that g belongs to the conjugacy class 3B and that the centralizer C_{E_{8}(C)}(g) has type E_{6}A_{2}. The precise structure of this centralizer may be found as follows. Since E_{8}(C) is simply connected, Theorem 8.2(3) implies that C_{E_{8}(C)}(g) is connected. Setting

α′ = α_{1} + α_{2} + 2α_{3} + 3α_{4} + 2α_{5} + α_{6},

we see that {α_{5}, α_{8}, α_{6}, α_{7}, α′, α_{2}} ∪ {α_{1}, α_{3}} is a basis of the root system of C_{E_{8}(C)}(g) (the bases of the components of type E_{6} and A_{2} have been ordered so that the numbering is consistent with [13, p. 250–251, 260–262]). From this we get an explicit homomorphism 3E_{6}(C) × SL_{3}(C) → E_{8}(C) onto the centralizer C_{E_{8}(C)}(g). The kernel is given by ⟨(z, ω^{2}I_{3})⟩, where z ∈ 3E_{6}(C) denotes the central element defined in Section 8.2. Thus C_{E_{8}(C)}(g) = 3E_{6}(C) ∘_{C_{3}} SL_{3}(C), and we denote elements in this central product by A · B where A ∈ 3E_{6}(C) and B ∈ SL_{3}(C). In particular we have g = z · I_{3} = 1 · ωI_{3}.

Now consider the subgroup E = ⟨z · I_{3}, x_{1} · β, y_{1} · γ⟩ which is seen to be an elementary abelian 3-group of rank 3 (here the elements x_{1}, y_{1} ∈ 3E_{6}(C) from Section 8.2 should not be confused with the elements x_{1}, y_{1} ∈ SL_{9}(C) from above). We have

C_{E_{8}(C)}(z · I_{3}, x_{1} · β) = C_{3E_{6}(C)∘_{C_{3}}SL_{3}(C)}(x_{1} · β) = ⟨y_{1} · γ, C_{3E_{6}(C)}(x_{1}) ∘_{C_{3}} C_{SL_{3}(C)}(β)⟩ .

We note that y_{1} · γ is not conjugate to its inverse in C_{E_{8}(C)}(z · I_{3}, x_{1} · β) since γ is not conjugate to γ^{−1} times a power of ωI_{3} in SL_{3}(C). Thus we have diag(1, 1, −1) ∉ W (E) and in particular W (E) ≠ GL_{3}(F_{3}). From the above we also get

C_{E_{8}(C)}(E) = ⟨y_{1} · γ, x_{1} · β, C_{3E_{6}(C)}(x_{1}, y_{1}) ∘_{C_{3}} C_{SL_{3}(C)}(β, γ)⟩
= ⟨y_{1} · γ, x_{1} · β, C_{3E_{6}(C)}(π^{−1}(E_{E_{6}}^{2b})) ∘_{C_{3}} Z(SL_{3}(C))⟩
= ⟨y_{1} · γ, x_{1} · β, (⟨z⟩ × G_{2}(C)) ∘_{C_{3}} Z(SL_{3}(C))⟩
= E × G_{2}(C),

using the computation of C_{3E_{6}(C)}(π^{−1}(E_{E_{6}}^{2b})) from the last part of the proof of Theorem 8.9. Since the preimage of E in 3E_{6}(C) × SL_{3}(C) is non-abelian it follows from Theorem 8.2(5) that E is non-toral in 3E_{6}(C) ∘_{C_{3}} SL_{3}(C). Now Theorem 8.2(1) shows that E is non-toral in E_{8}(C) (alternatively one could also just observe that C_{E_{8}(C)}(E) has rank less than 8). From what we have already proved we then see that E is conjugate to either E_{E_{8}}^{3a} or E_{E_{8}}^{3b} in E_{8}(C). Since we already know C_{E_{8}(C)}(E_{E_{8}}^{3a}) we conclude that E must be conjugate to E_{E_{8}}^{3b} (alternatively one could also compute the class distribution of E directly). In particular we have C_{E_{8}(C)}(E_{E_{8}}^{3b}) = E_{E_{8}}^{3b} × G_{2}(C) and W (E_{E_{8}}^{3b}) ≠ GL_{3}(F_{3}).

Using the inclusion 3E_{6}(C) ⊆ 3E_{6}(C) ∘_{C_{3}} SL_{3}(C) ⊆ E_{8}(C) we may also consider the subgroup E_{3E_{6}}^{4} ⊆ 3E_{6}(C) from Theorem 8.7 as a subgroup of E_{8}(C). Since E_{3E_{6}}^{4} is nontoral in 3E_{6}(C), it is also non-toral in 3E_{6}(C) ∘_{C_{3}} SL_{3}(C), and hence also in E_{8}(C) by Theorem 8.2(1). Thus E_{3E_{6}}^{4} must be conjugate in E_{8}(C) to one of the groups E_{E_{8}}^{4a}, E_{E_{8}}^{4b} or E_{E_{8}}^{4c}. Comparing with the class distributions we can rule out E_{E_{8}}^{4a} and E_{E_{8}}^{4b}, so we conclude that E_{3E_{6}}^{4} is conjugate to E_{E_{8}}^{4c}. From Theorem 8.7 we have C_{3E_{6}(C)}(E_{3E_{6}}^{4}) = E_{3E_{6}}^{4}. Hence C_{E_{8}(C)}(E_{3E_{6}}^{4}) = E_{3E_{6}}^{4} ∘_{C_{3}} SL_{3}(C) from which we get C_{E_{8}(C)}(E_{E_{8}}^{4c}) = E_{E_{8}}^{4c} ∘_{C_{3}} SL_{3}(C). We determine the precise structure of the central product below after the computation of W (E_{E_{8}}^{4c}).

Exact Weyl groups: Recall from above that E_{E_{8}}^{3a} is conjugate to ⟨\overline{x_{2}}, \overline{y_{2}}, \overline{a}⟩. If W (E_{E_{8}}^{3a}) was larger than the group W′(E_{E_{8}}^{3a}) from above, we then see that W (E_{E_{8}}^{3a}) would have to contain one of the groups

[ \begin{matrix}GL_{2}(F_{3}) & \begin{matrix}∗ \\ ∗ \end{matrix} \\ 0 0 & ε \end{matrix} ] or SL_{3}(F_{3})

which are the minimal overgroups of W′(E_{E_{8}}^{3a}) inside GL_{3}(F_{3}). Thus W (E_{E_{8}}^{3a}) would have to contain one of the matrices diag(1, 2, 1) or I_{3} + e_{3,2}. This would mean that inside C_{E_{8}(C)}(\overline{x_{2}}, \overline{a}) there would be an element which conjugates \overline{y_{2}} into either \overline{y_{2}}^{2} or \overline{y_{2}a}. However from above we have

C_{E_{8}(C)}(\overline{x_{2}}, \overline{a}) = \overline{⟨y_{2}, \{A ⊕ B ⊕ C | det ABC = 1\}⟩},

and from this it is easily seen that no such element exists. Thus W (E_{E_{8}}^{3a}) = W′(E_{E_{8}}^{3a}) as claimed. For the group E_{E_{8}}^{3b} we have SL_{3}(F_{3}) ⊆ W (E_{E_{8}}^{3b}) ≠ GL_{3}(F_{3}) and hence W (E_{E_{8}}^{3b}) = SL_{3}(F_{3}).

As in the proof of Theorem 8.9 we show that the remaining Weyl groups equal the lower bounds already established, by looking at what this would imply for the subgroups E_{E_{8}}^{3a} and E_{E_{8}}^{3b}. For E = E_{E_{8}}^{4a}, E_{E_{8}}^{4b}, E_{E_{8}}^{5a}, and E_{E_{8}}^{5b}, any proper overgroup of W (E) contains an element which normalizes E_{E_{8}}^{3a} but induces an automorphism on it not contained in its Weyl group. For E_{E_{8}}^{4c} the result follows by considering the subgroup E_{E_{8}}^{3b}.

It remains only to determine the precise structure of the central product C_{E_{8}(C)}(E_{E_{8}}^{4c}) = E_{E_{8}}^{4c} ∘_{C_{3}} SL_{3}(C). From the structure of W (E_{E_{8}}^{4c}) we see that the subgroup ⟨\overline{x_{2}}⟩ is invariant under the action of W (E_{E_{8}}^{4c}). Thus a conjugates which sends to E_{E_{8}}^{4c} to E_{3E_{6}}^{4} we see that ⟨\overline{x_{2}}⟩ must be send to a W (E_{3E_{6}}^{4})-invariant subgroup of E_{3E_{6}}^{4}. From the structure of W (E_{3E_{6}}^{4}) we see that there is only one such subgroup, namely ⟨z · I_{3}⟩ = ⟨1 · ωI_{3}⟩. As this is exactly the center of the SL_{3}(C)-component of C_{E_{8}(C)}(E_{3E_{6}}^{4}) = E_{3E_{6}}^{4} ∘_{C_{3}} SL_{3}(C), we see that C_{E_{8}(C)}(E_{E_{8}}^{4c}) = E_{E_{8}}^{4c} ∘_{⟨\overline{x_{2}}⟩} SL_{3}(C). □

8.4. The group 2E_{7}(C), p = 3. In this section we consider the elementary abelian 3subgroups of 2E_{7}(C). We let H be a maximal torus of 2E_{7}(C), Φ(E_{7}) be the root system relative to H, and choose a realization ([118, p. 133]) (u_{α})_{α∈Φ(E_{7})} of Φ(E_{7}) in 2E_{7}(C). By [118, 8.1.4(iv)] we may suppose that the root subgroups (u′_{α})_{α∈Φ(E_{6})} in 3E_{6}(C) ⊆ 2E_{7}(C) coming from the choice of root subgroups for 3E_{6}(C) from Section 8.2 satisfy u_{α} = u′_{α} for α ∈ Φ(E_{6}).

For α = α_{i}, 1 ≤ i ≤ 7, and t ∈ C^{×} we define the elements

n_{α}(t) = u_{α}(t)u_{−α}(−1/t)u_{α}(t), h_{α}(t) = n_{α}(t)n_{α}(1)^{−1}.

Then the maximal torus consists of the elements ∏_{i=1}^{7} h_{α_{i}}(t_{i}) and the normalizer N(H) of the maximal torus is generated by H and the elements n_{i} = n_{α_{i}}(1), 1 ≤ i ≤ 7.

As in Section 8.2 we define the following elements in 3E_{6}(C) ⊆ 2E_{7}(C):

z = h_{α_{1}}(ω)h_{α_{3}}(ω^{2})h_{α_{5}}(ω)h_{α_{6}}(ω^{2}),  a = h_{α_{1}}(ω)h_{α_{3}}(ω^{2})h_{α_{5}}(ω^{2})h_{α_{6}}(ω),
x_{2} = h_{α_{2}}(ω^{2})h_{α_{3}}(ω^{2})h_{α_{5}}(ω^{2}),
y_{2} = n_{1}n_{2}n_{3}n_{4}n_{3}n_{1}n_{5}n_{4}n_{2}n_{3}n_{4}n_{5}n_{6}n_{5}n_{4}n_{2}n_{3}n_{1}n_{4}n_{3}n_{5}n_{4}n_{6}n_{5}h_{α_{2}}(−1).

Notation 8.17. The conjugacy classes of elements of order 3 in 2E_{7}(C) are given in [63, Table VI] and [30, Table 6] from which we take our notation. In particular, there are 5 such conjugacy classes, which we label 3A, 3B, 3C, 3D and 3E. Moreover these classes may be distinguished by their traces on 𝔢_{7}, except for the classes 3A and 3D which have the same trace. Since the trace of the element h ∈ H is given by 7 + ∑_{α∈Φ(E_{7})} α(h) we easily obtain the inclusions

3C[3E_{6}] ⊆ 3C[2E_{7}],  3E[3E_{6}] ⊆ 3B[2E_{7}],  3E′[3E_{6}] ⊆ 3B[2E_{7}],

corresponding to the inclusion 3E_{6}(C) ⊆ 2E_{7}(C).

Theorem 8.18. The conjugacy classes of non-toral elementary abelian 3-subgroups of 2E_{7}(C) are given by the following table.

rank name

ordered basis 2E_{7}(C)-class dist. C_{2E_{7}(C)}(E) Z(C_{2E_{7}(C)}(E))
3 E_{2E_{7}}^{3}  ⟨a, x_{2}, y_{2}⟩  3C^{26}  E_{2E_{7}}^{3} × SL_{2}(C) E_{2E_{7}}^{3} × Z(2E_{7}(C))
4 E_{2E_{7}}^{4} ⟨z, a, x_{2}, y_{2}⟩  3B^{2}3C^{78}  E_{2E_{7}}^{4} ∘_{⟨z⟩} T_{1}  E_{2E_{7}}^{4} ∘_{⟨z⟩} T_{1}

In particular we have _{3}Z(C_{2E_{7}(C)}(E)) = E for any non-toral elementary abelian 3-subgroup of 2E_{7}(C).

The Weyl groups of these groups with respect to the given ordered bases are given as follows:

W (E_{2E_{7}}^{3}) = SL_{3}(F_{3}), W (E_{2E_{7}}^{4}) = [ \begin{matrix}ε & ∗ ∗ ∗ \\ 0 0 0 & SL_{3}(F_{3}) \end{matrix} ]

Remark 8.19. Note that our information on the rank 3 subgroup E_{2E_{7}}^{3} corrects [63].

Proof of Theorem 8.18. Non-toral subgroups: From the way the realization (u_{α})_{α∈Φ(E_{7})} is chosen above, it follows from Theorem 8.7 that E_{2E_{7}}^{3} and E_{2E_{7}}^{4} are elementary abelian 3-subgroups of 2E_{7}(C) and that we have

W (E_{2E_{7}}^{3}) ⊇ SL_{3}(F_{3}), W (E_{2E_{7}}^{4}) ⊇ [ \begin{matrix}1 & ∗ ∗ ∗ \\ 0 0 0 & SL_{3}(F_{3}) \end{matrix} ]

In particular we see that both W (E_{2E_{7}}^{3}) and W (E_{2E_{7}}^{4}) have orders divisible by 13 and since 13 ∤ |W (E_{7})| = 2^{10} · 3^{4} · 5 · 7, we conclude by Theorem 8.2(2) that E_{2E_{7}}^{3} and E_{2E_{7}}^{4} are non-toral in 2E_{7}(C). By [63, Thm. 11.16] we know that there are precisely two conjugacy classes of non-toral elementary abelian 3-subgroups in 2E_{7}(C), and thus E_{2E_{7}}^{3} and E_{2E_{7}}^{4} represent these two conjugacy classes.

Class distributions: The class distributions follows directly from the class distributions of the groups E_{3E_{6}}^{3} and E_{3E_{6}}^{4} given in Theorem 8.7 and the information in 8.17 about the behavior of conjugacy classes in 3E_{6}(C) under the inclusion 3E_{6}(C) ⊆ 2E_{7}(C).

Weyl groups: Using our realization (u_{α})_{α∈Φ(E_{7})} we may define a canonical map ϕ : W → N(H) as follows ([118, 9.3.3]): If w = s_{α_{i_{1}}} … s_{α_{ir}} is a reduced expression for w ∈ W we let ϕ(w) = n_{i_{1}} … n_{i_{r}} (by [118, 8.3.3 and 9.3.2] this does not depend on the reduced expression for w). Note that the element ϕ(w) is a representative in N(H) for w ∈ W. Now let w_{0} ∈ W be the longest element in W, and let n_{0} = ϕ(w_{0}). From [13, p. 264–266] it follows that w_{0} equals the scalar transformation −1, and so conjugation by n_{0} acts as inversion on H. Now let w ∈ W and define w′ by ww′ = w_{0}. Since w_{0} is central in our case, we have (ww′) w^{−1} = w^{−1} (ww′) = w′ so we conclude that w′w = ww′ = w_{0}. Now let ℓ be the length function on W. By [71, p. 16] we have ℓ(w) + ℓ(w′) = ℓ(w_{0}). In general the map ϕ is not a homomorphism, but we do have ϕ(w_{1}w_{2}) = ϕ(w_{1})ϕ(w_{2}) if ℓ(w_{1}w_{2}) = ℓ(w_{1}) + ℓ(w_{2}) by [118, 9.3.4(i)]. From this it follows that ϕ(w)ϕ(w′) = ϕ(w′)ϕ(w) = ϕ(w_{0}) = n_{0}, and we conclude that n_{0} commutes with ϕ(w) for all w ∈ W.

Now consider the element

w = s_{1}s_{2}s_{3}s_{4}s_{3}s_{1}s_{5}s_{4}s_{2}s_{3}s_{4}s_{5}s_{6}s_{5}s_{4}s_{2}s_{3}s_{1}s_{4}s_{3}s_{5}s_{4}s_{6}s_{5}.

Using the fact that the length of an element is given by the number of positive roots it sends to negative roots ([71, Cor. 1.7]), we see that the above product is a reduced expression for w. Thus we have y_{2} = ϕ(w)h_{α_{2}}(−1). From the above we then conclude that conjugation by n_{0} acts as follows:

z ↦ z^{2},  a ↦ a^{2},  x_{2} ↦ x_{2}^{2},  y_{2} ↦ y_{2}.

Thus n_{0} normalizes E_{2E_{7}}^{4} and gives the element diag(2, 2, 2, 1) in W (E_{2E_{7}}^{4}). Combined with the above we conclude that

W (E_{2E_{7}}^{4}) ⊇ [ \begin{matrix}ε & ∗ ∗ ∗ \\ 0 0 0 & SL_{3}(F_{3}) \end{matrix} ]

From the inclusion Φ(E_{7}) ⊆ Φ(E_{8}) we get the inclusion 2E_{7}(C) ⊆ E_{8}(C), so we may consider E_{2E_{7}}^{3} and E_{2E_{7}}^{4} as subgroups of E_{8}(C) as well. Since the orders of their Weyl groups in 2E_{7}(C) are divisible by 13 and 13 ∤ |W (E_{8})| = 2^{14} · 3^{5} · 5^{2} · 7 we see from Theorem 8.2(2) that E_{2E_{7}}^{3} and E_{2E_{7}}^{4} remain non-toral in E_{8}(C). Using Theorem 8.15 and the class distributions from above we conclude that E_{2E_{7}}^{3} and E_{2E_{7}}^{4} are conjugate to E_{E_{8}}^{3b} and E_{E_{8}}^{4c} respectively in E_{8}(C). Theorem 8.15 now shows that the lower bounds found above are indeed the Weyl groups of E_{2E_{7}}^{3} and E_{2E_{7}}^{4} in 2E_{7}(C).

Centralizers: For the computation of the centralizer of E_{2E_{7}}^{3} we consider the element g = h_{α_{1}}(ω)h_{α_{3}}(ω^{2}) ∈ 2E_{7}(C). By using [30, Table 6] we see that g belongs to the conjugacy class 3B and that the centralizer C_{2E_{7}(C)}(g) has type A_{5}A_{2}. The precise structure of this centralizer may be found as follows. Since 2E_{7}(C) is simply connected, Theorem 8.2(3) implies that C_{2E_{7}(C)}(g) is connected. Setting

α′ = α_{1} + α_{2} + 2α_{3} + 3α_{4} + 2α_{5} + α_{6},

we see that {α_{5}, α_{6}, α_{7}, α′, α_{2}} ∪ {α_{1}, α_{3}} is a basis of the root system of C_{2E_{7}(C)}(g) (the bases of the components of type A_{5} and A_{2} have been ordered so that the numbering is consistent with [13, p. 250–251]). From this we get an explicit homomorphism SL_{6}(C) × SL_{3}(C) → 2E_{7}(C) onto the centralizer C_{2E_{7}(C)}(g). The kernel is given by ⟨(ωI_{6}, ω^{2}I_{3})⟩.

Thus C_{2E_{7}(C)}(g) = SL_{6}(C) ∘_{C_{3}} SL_{3}(C), and we denote elements in this central product by A · B where A ∈ SL_{6}(C) and B ∈ SL_{3}(C). In particular we have g = ωI_{6} · I_{3} = I_{6} · ωI_{3}.

Now consider the subgroup E = ⟨ωI_{6} · I_{3}, (β ⊕ β) · β, (γ ⊕ γ) · γ^{2}⟩ which is seen to be an elementary abelian 3-group of rank 3. We have

C_{2E_{7}(C)}(ωI_{6} · I_{3}, (β ⊕ β) · β) = C_{SL_{6}(C)∘_{C_{3}}SL_{3}(C)}((β ⊕ β) · β)
= ⟨(γ ⊕ γ) · γ^{2}, C_{SL_{6}(C)}(β ⊕ β) ∘_{C_{3}} C_{SL_{3}(C)}(β)⟩ .

From this we get

C_{2E_{7}(C)}(E) = ⟨(γ ⊕ γ) · γ^{2}, (β ⊕ β) · β, C_{SL_{6}(C)}(β ⊕ β, γ ⊕ γ) ∘_{C_{3}} C_{SL_{3}(C)}(β, γ)⟩
= ⟨(γ ⊕ γ) · γ^{2}, (β ⊕ β) · β, C_{SL_{6}(C)}(β ⊕ β, γ ⊕ γ) ∘_{C_{3}} Z(SL_{3}(C))⟩ .

Here C_{SL_{6}(C)}(β⊕β, γ⊕γ) = Δ_{2,3}({A ∈ GL_{2}(C)|(det A)^{3} = 1}) is generated by Δ_{2,3}(ω^{2}I_{2}) = ω^{2}I_{6} and Δ_{2,3}(SL_{2}(C)). From this we get

C_{2E_{7}(C)}(E) = ⟨E, Δ_{2,3}(SL_{2}(C))⟩ ≅ E × SL_{2}(C).

Since the preimage of E in SL_{6}(C) × SL_{3}(C) is non-abelian it follows from Theorem 8.2(5) that E is non-toral in SL_{6}(C) ∘_{C_{3}} SL_{3}(C). Now Theorem 8.2(1) shows that E is nontoral in 2E_{7}(C) (alternatively one could also just observe that C_{2E_{7}(C)}(E) has rank less than 7). It then follows that E is conjugate to E_{2E_{7}}^{3} in 2E_{7}(C). In particular we have C_{2E_{7}(C)}(E_{2E_{7}}^{3}) = E_{2E_{7}}^{3} × SL_{2}(C). Hence Z(C_{2E_{7}(C)}(E_{2E_{7}}^{3})) = E_{2E_{7}}^{3} × Z(2E_{7}(C)) since the center of 2E_{7}(C) has order 2.

To compute the centralizer of E_{2E_{7}}^{4} we note that C_{2E_{7}(C)}(z) has centralizer type E_{6}T_{1}, and that the E_{6} component corresponds to the subgroup 3E_{6}(C) ⊆ 2E_{7}(C). A computation shows that the T_{1} component is given by T_{1} = {h(t^{2}, t^{3}, t^{4}, t^{6}, t^{5}, t^{4}, t^{3})|t ∈ C^{×}}, and thus we get C_{2E_{7}(C)}(z) = 3E_{6}(C) ∘_{⟨z⟩} T_{1}. Theorem 8.7 now shows that C_{2E_{7}(C)}(E_{2E_{7}}^{4}) = C_{3E_{6}(C)}(E_{2E_{7}}^{4}) ∘_{⟨z⟩} T_{1} = E_{2E_{7}}^{4} ∘_{⟨z⟩} T_{1}. □

9. Non-toral elementary abelian p-subgroups of projective unitary groups

The purpose of this short section is to describe the non-toral elementary abelian subgroups of PGL_{n}(C), which by Theorem 8.4 is equivalent to finding them for its compact form P U(n), as well as give information about centralizers and Weyl groups. The subgroups are easily determined and are described in [63, §3]—we here just add some extra information centralizers and Weyl groups which we need in our proof of Theorem 1.1.

We first introduce a useful subgroup. Suppose n = p^{r}k and consider the extraspecial group p_{+}^{1+2r} embedded in GL_{n}(C) by taking k copies of one of the p − 1 faithful irreducible p^{r}-dimensional representations. (They all have the same image; see [72, Satz 16.14].) Note that this embedding maps the center of p_{+}^{1+2r} to the elements of order p in the center of GL_{n}(C). Let Γ_{r} denote the subgroup of GL_{n}(C) given by the subgroup generated by the image of p_{+}^{1+2r} and the center of GL_{n}(C). Note that as an abstract group Γ_{r} fits into an extension sequence

1 → C^{×} → Γ_{r} → Γ̄_{r} → 1

where C^{×} identifies with the center of C^{×} and Γ̄_{r} ≅ (Z/p)^{2r} identifies with the image of Γ_{r} in PGL_{n}(C). (The matrices for Γ_{r} are written explicitly for k = 1 in [104, p. 56] where it is called Γ_{p^{r}}^{U}.)

Theorem 9.1. Suppose E is a non-toral elementary p-subgroup of PGL_{n}(C) (any prime p). Then, up to conjugacy E can be written E = Γ̄_{r} × Ā, for some r ≥ 1 and some abelian subgroup A of C_{GL_{n}(C)}(Γ_{r}) ≅ GL_{k}(C).

For a given r the number of conjugacy classes of such subgroups E are in one-to-one correspondence with conjugacy classes of toral elementary abelian p-subgroups Ā of PGL_{k}(C) ≅ C_{PGL_{n}(C)}(Γ̄_{r})_{1} (allowing the trivial subgroup), and the centralizer of E is given by C_{PGL_{n}(C)}(E) ≅ Γ̄_{r} × C_{PGL_{k}(C)}(Ā).

The Weyl group equals

W_{PGL_{n}(C)}(E) = [ \begin{matrix}Sp(Γ̄_{r}) & 0 \\ ∗ & W_{PGL_{k}(C)}(Ā) \end{matrix} ]

Here Sp(Γ̄_{r}) is the symplectic group relative to the symplectic product coming from the commutator product [·, ·] : Γ̄_{r} × Γ̄_{r} → Z/p ⊆ C^{×} and the symbol ∗ denotes a rank(Ā) × 2r matrix with arbitrary entries.

An element α ∈ Sp(Γ̄_{r}) ⊆ W_{PGL_{n}(C)}(E) acts as up to conjugacy as α×1 on C_{PGL_{n}(C)}(E) ≅ Γ̄_{r} × C_{PGL_{k}(C)}(Ā).

Sketch of proof of Theorem 9.1. The existence of the decomposition E = Γ̄_{r} × Ā follows from Griess [63, Thm. 3.1] and the statements about uniqueness follows by representation theory of the extraspecial p-groups.

Since the image of p_{+}^{1+2r} is the sum of k identical irreducible representations we have by Schur’s lemma that C_{GL_{n}(C)}(Γ_{r}) ≅ GL_{k}(C) (see also [104, Prop. 4]). From this the centralizer in PGL_{n}(C) can easily be worked out.

In the case where Ā is trivial the statement about Weyl groups is given in [104, Thm. 6] (and just uses elementary character theory). The general case follows similarly, again using character theory.

For the statement about the Weyl group action, first note that Out(Γ̄_{r} × PGL_{k}(C)) ≅ Aut(Γ̄_{r}) × Out(PGL_{k}(C)). An element α ∈ Sp(Γ_{r}) = W_{PGL_{n}(C)}(Γ̄_{r}) acts as an inner automorphism on PGL_{k}(C) since this is true for the action on C_{GL_{n}(C)}(Γ_{r}) ≅ GL_{k}(C) by character theory. Hence we can choose a representative g ∈ N_{PGL_{n}(C)}(Γ̄_{r}) of α which acts as α × 1 on C_{PGL_{n}(C)}(Γ̄_{r}) ≅ Γ̄_{r} × PGL_{k}(C). Hence g is also a representative of α ∈ Sp(Γ̄_{r}) ⊆ W_{PGL_{n}(C)}(Γ̄_{r} × Ā). The claim now follows. □

10. Calculation of the obstruction groups

In this section we show that the existence and uniqueness obstructions to lifting our diagram in the homotopy category to a diagram in the category of spaces identically vanish. More precisely, we will show the following theorem.

Theorem 10.1. Suppose that X is any of the following p-compact groups F_{4}_{3}^{∧}, E_{6}_{3}^{∧}, E_{7}_{3}^{∧}, E_{8}_{3}^{∧}, E_{8}_{5}^{∧} or P U(n)_{p}^{∧} (any p), or suppose that p is odd and X is connected and center-free with H^{∗}(BX; Z_{p}) a polynomial algebra. Then

lim_{A(X)}^{i}π_{j}(BZ(C_{X}(−))) = 0, for all i, j.

(See Theorem 12.2 for an explanation of why exactly these p-compact groups need attention.) Note that for the purpose of Theorem 1.4 we only need to calculate the above groups for j = 1, 2 and i = j or i = j + 1.

We prove the theorem by filtering the functor F_{j} = π_{j}(BZ(C_{X}(−))), and showing that all filtration quotients vanish (with a small twist for P U(2)_{2}^{∧}). First we show that the quotient functor of F_{j} concentrated on all the toral elementary abelian p-subgroups has vanishing limits, using a Mackey functor argument which first appeared in [47]. This takes care of the case where H^{∗}(BX; Z_{p}) is a polynomial algebra since in this case all subgroups are toral by Lemma 7.8. For the exceptional compact Lie groups we then continue and filter the non-toral part of the functor by functors concentrated on only one non-toral subgroup, and use a formula of Oliver [103] to show that the higher limits of these subquotient functors all vanish. For P U(n) we use a variant of this technique by suitably grouping the non-toral subgroups and using a combination of Oliver’s formula and the Mackey functor argument we used for the toral part.

We use the notation St_{G} to denote the Steinberg module over Z_{p} of a finite group of Lie type G of characteristic p, defined as the top homology group with Z_{p} coefficients of the Tits building of G (see e.g., [69]). In the special case of GL(E) we also write St(E) for the Steinberg module.

10.1. The toral part. Define a quotient functor F_{j}^{tor} of F_{j} by setting F_{j}^{tor}(V) = F_{j}(V) if V is toral and F_{j}^{tor}(V) = 0 if V is non-toral. Let A^{tor}(X) denote the full subcategory of A(X) consisting of toral subgroups. From the chain complex defining higher limits (see e.g., [61, 3.3]) it follows that

lim_{A(X)}^{∗}F_{j}^{tor} ≅ lim_{A^{tor}(X)}^{∗}F_{j}^{tor}

In order to use a Mackey functor argument on the right-hand side we need a more explicit description of the functor F_{j}^{tor}.

Lemma 10.2. Fix a connected p-compact group X and let T̆ be a discrete approximation to a maximal torus T in X. For a non-trivial elementary abelian p-subgroup V ⊆ T̆, let W_{X}(V) denote the Weyl group of C_{X}(V) and let W_{X}(V)_{1} denote the Weyl group of C_{X}(V)_{1} (see [49, Thm. 7.6]).

If T̆^{W_{X}(V)_{1}} is a discrete approximation to Z(Y_{1}) then T̆^{W_{X}(V)} is a discrete approximation to Z(C_{X}(V)). In particular in this case π_{1}(BZ(C_{X}(V))) = H^{1}(W_{X}(V); L_{X}) and π_{2}(BZ(C_{X}(V)) = (L_{X})^{W_{X}(V)}, where L_{X} = π_{1}(T).

Remark 10.3. For a connected p-compact group X and p odd, the fixed points T̆^{W_{X}} always equals a discrete approximation to the center of X by [49, Thm. 7.6]. If X is the p-completion of a connected compact Lie group then this is likewise the case for p = 2 unless X contains a direct factor isomorphic to SO(2n + 1)_{2}^{∧}, by [83, Thm. 1.6].

Proof of Lemma 10.2. Set Y = C_{X}(V) and π = π_{0}(Y) for short. Since V is toral, T̆ is in a canonical way a discrete approximation to a maximal torus in Y.

First observe that the center of Y has discrete approximation in T̆. Indeed, otherwise there would by [49, Thm. 6.4] exist a central homomorphism f : Z/p^{n} → N̆_{p,Y} with image not in T̆, which would produce a homomorphism f′ : Z/p^{n} → N̆_{p,X} commuting with T̆ but not in T̆, which contradicts the fact that T is self-centralizing in X by [48, Thm. 9.1], since X is connected.

Suppose that T̆^{W_{X}(V)_{1}} is a discrete approximation to Z(Y_{1}) and set C = T̆^{W_{X}(V)}. We want to show that C is central in Y. Let f : BC → BY_{1} be the natural inclusion. We have an obvious diagram with horizontal maps fibrations

[figure: map(BC, BY_{1})_{{f}}; map(BC, BY)_{f}; map(BC, Bπ)_{0}; BY_{1}; BY; Bπ]

where {f} denotes the set of homotopy classes of maps BC → BY_{1} generated by f under the π-action on BY_{1}. If we can show that {f} consists of just f then it follows from the five-lemma that the middle vertical map is a homotopy equivalence, since our assumption implies that C is central in Y_{1}.

To see that the action is trivial consider the following diagram:

\xymatrix{ & BN_{Y_{1}} \ar[r]^{g̃} \ar[d] & BN_{Y_{1}} \ar[d] \\ BC \ar[r]^{f} \ar[ur]^{f̃} & BY_{1} \ar[r]^{g} & BY_{1}}

where f̃ is the natural inclusion of BC in BN_{Y_{1}}, g is an element in Aut(BY_{1}) induced by an element in π and g̃ is the corresponding self-map of BN_{Y_{1}} defined via Lemma 2.1. However by the definition of C, the composite g̃f̃ homotopic to f̃ for all g induced by an element in π, so f is homotopic to gf as well. Hence we have shown that C is central in Y and since the center of Y has discrete approximation in T̆ it is obviously the largest subgroup with this property. So C is a discrete approximation to ZY as wanted.

The last statement about the homotopy groups now follows easily using the long exact sequence in group cohomology. □

Remark 10.4. The above lemma should be compared to Lemma 4.5 and Remark 4.6 which have slightly different assumptions and conclusions.

The following lemma is essentially contained in [47, §8].

Lemma 10.5. Let X be a connected p-compact group, and assume that for each non-trivial toral elementary abelian p-subgroup V ∈ A(X) the fixed point set T̆^{W_{X}(V)_{1}} is a discrete approximation to Z(C_{X}(V)_{1}). Then

lim_{A(X)}^{i}F_{j}^{tor} = \{ \begin{matrix}H^{2−j}(W_{X}; L_{X}) & if i = 0 and j = 1, 2 \\ 0 & otherwise. \end{matrix}

where H^{2−j}(W_{X}; L_{X}) ≅ π_{j}(BZ(X)) if T̆^{W} is a discrete approximation to Z(X).

In particular, if p is odd, or more generally if for all reflections s ∈ W_{X} the singular set σ(s) equals the fixed point set T̆^{⟨s⟩} then the assumptions above are satisfied and lim_{A(X)} ^{i}F_{j}^{tor} = π_{j}(BZ(X)) if i = 0 and zero otherwise. (See [49, Def. 7.3] and Remark 10.6 for the definition of σ(s).)

Proof. The first part of the proof consists of a translation of [47, §8] into the current notation. By [50, Prop. 3.4] all morphism in A(X) between toral subgroups V → X and V′ → X are induced by inclusion and action by W_{X}. Hence we can identify A^{tor}(X), up to equivalence of categories with a category which has objects non-trivial subgroups of _{p}T̆ ≅ (Z/p)^{r} (where r is the rank of T) and morphisms the homomorphism between subgroups induced by inclusions and action by W_{X}. Also, by [49, Thm. 7.6], W_{X}(V) consists of the elements in W_{X} which pointwise fixes V. Hence Lemma 10.2 shows that the functor F_{2}^{tor} on A^{tor}(X) is isomorphic to the functor α_{Γ,M}^{0} on A_{Γ} from [47, §8], where Γ = W_{X} and M = L_{X}. Likewise F_{1}^{tor} is isomorphic to α_{Γ,M}^{1}. (Note that there is the slight difference from [47, §8] that M is a Z_{p}Γ-module rather than an F_{p}Γ-module, but this makes no difference.) Therefore [47, §8] (which is a Mackey functor argument, which can also be deduced from [46] or [73]) implies the first part of the lemma about obstruction groups.

To see the last part about the singular set recall that for an abelian subgroup A ⊆ T̆ we have by [49, Thm. 7.6] that

⋂ ; reflections s ∈ W_{X} such that A ⊆ σ(s) σ(s)

is a discrete approximation to Z(C_{X}(A)_{1}). Hence if σ(s) equals T̆^{⟨s⟩} then the assumptions of the first part are obviously satisfied since (again by [49, Thm. 7.6]) W_{X}(A)_{1} is generated by reflections s ∈ W_{X} with A ⊆ σ(s). □

Remark 10.6. Let G be a compact connected Lie group with maximal torus T, and let α be a root of G relative to T with corresponding reflection s_{α}. In this case the singular set σ(s_{α}) is just the discrete approximation of the kernel U_{α} of α on T. (To see this note that by [15, §4, no. 5] the reflection s_{α} lift to an element n_{α} (denoted by ν(θ) in [15]) which satisfies n_{α}^{2} = exp(α^{∨}/2); the statement now follows—cf. [83, Pf. of Prop. 3.1(ii)].)

Explicit calculations [83, Prop. 3.1(ii)] (see also [55], [75, Prop. 3.2(vi)], and [106, §4]) show that for a compact connected Lie group G, σ(s) in fact always equal to T̆^{⟨s⟩} except when G contains a direct factor isomorphic to SO(2n+1). Combining this with Lemma 10.5, now gives the following calculation of the toral part of the obstruction groups, whose full strength at p = 2 we will however not use here.

Corollary 10.7. Let G be a compact connected Lie group with no direct factors isomorphic to SO(2n + 1) and p a prime. Set X = G_{p}^{∧}. Then

lim_{A(X)}^{i}F_{j}^{tor} = \{ \begin{matrix}π_{j}(BZ(X)) & if i = 0 \\ 0 & otherwise. \end{matrix}

□

Proof of Theorem 10.1 when H^{∗}(BX; Z_{p}) is polynomial, p odd. If H^{∗}(BX; Z_{p}) is a polynomial algebra concentrated in even degrees then all elementary abelian p-subgroups are toral by Lemma 7.8, so F = F^{tor}. Since p is odd the assumptions of Lemma 10.5 are satisfied and Theorem 10.1 follows. □

10.2. The non-toral part for the exceptional groups. In this subsection we prove Theorem 10.1 when X is the p-completion of one of the exceptional groups and p odd. Let F_{j}^{E} denote the subquotient functor of F_{j} concentrated on a non-toral elementary abelian p-group E. By Oliver’s formula [103, Prop. 4]

lim_{A(X)}^{i}F_{j}^{E} = \{ \begin{matrix}Hom_{W} (St(E), F_{j}(E)) & if i = rk E − 1 \\ 0 & otherwise. \end{matrix}

We now embark on proving some lemmas which will be used to show that these obstructions groups identically vanish. (For the use in Theorem 1.4 we actually only need this when E has rank at most four.)

Since ZC_{X}(E) is the p-completion of an abelian compact Lie group (by [49, Thm. 1.1]), F_{j} = 0 unless j = 1, 2. The following lemma reduces the problem of showing that the obstruction groups vanish to showing that Hom_{W (E)}(St(E),_{p}ZC_{X}E) = 0, where _{p}ZC_{X}E is the finite group of elements of order p.

Lemma 10.8. Let A be an abelian compact Lie group, and let _{p}A and A^{p} denote the kernel and the image of the pth power map on A (using multiplicative notation). Let P be a finitely generated projective Z_{p}W -module for a finite group W, and assume that A has a module action of W.

Then Hom_{W} (P,_{p}A) = 0 if and only if Hom_{W} (P, π_{1}(A)⊗Z_{p}) = Hom_{W} (P, π_{0}(A)⊗Z_{p}) = 0.

Proof. The long exact sequence of homotopy groups associated to the exact sequence of groups 1 → A^{p} → A → A/A^{p} → 1 shows that the inclusion A^{p} ↪ A induces an isomorphism π_{1}(A^{p}) →^{≃} π_{1}(A) and an injection π_{0}(A^{p}) ↪ π_{0}(A).

Hence the exact sequence 1 → _{p}A → A →^{p} A^{p} → 1 produces the following diagram, where the row, as well as the sequence going through π_{i}(A) instead of π_{i}(A^{p}), is exact.

\xymatrix{π_{1}(A) \ar[r]^{p} \ar[dr]^{p} & π_{1}(A^{p}) \ar[d]^{≃} \ar[r] & π_{0}(_{p}A) \ar[r] & π_{0}(A) \ar[r]^{p} \ar[dr]^{p} & π_{0}(A^{p}) \ar@{^{(}->}[d] \\  & π_{1}(A) &  &  & π_{0}(A)}

Apply the exact functor Hom_{W} (P, − ⊗ Z_{p}) to this diagram. The lemma now follows from Nakayama’s lemma, using that π_{0}(A) is finite and π_{1}(A) is finitely generated. □

The following elementary observation is so useful that it is worth stating explicitly.

Lemma 10.9. Suppose that W is a subgroup of GL(E), p odd, such that −1 ∈ W. Then Hom_{W} (St(E), E) = 0.

Proof. Set Z = ⟨−1⟩. Since Z acts trivially on St(E) we have

Hom_{W} (St(E), E) ⊆ Hom_{Z}(St(E), E) = Hom_{Z}(St(E), E^{Z}) = 0.
□

We also need the following lemma, which is a special case of a theorem of Steve Smith [117].

Lemma 10.10. Let G be a finite group of Lie type of characteristic p, and let P be a parabolic subgroup of G with corresponding unipotent radical U and Levi subgroup L ≅ P/U. Suppose that W is a subgroup U ⊆ W ⊆ P, and let M be an F_{p}W -module.

(1) If U acts trivially on M, then Hom_{W} (St_{G}, M) = Hom_{W/U}(St_{L}, M).

(2) If St_{L} ⊗F_{p} is irreducible as an F_{p}W/U-module and if M has a finite filtration as a F_{p}W -module, with filtration quotients of F_{p}-dimension strictly less than dim_{Z_{p}} St_{L} then Hom_{W} (St_{G}, M) = 0.

Proof. Since U acts trivially on M, Hom_{W} (St_{G}, M) = Hom_{W/U}((St_{G})_{U}, M) where (−)_{U} denotes coinvariants. But since the Steinberg module is self-dual, as is clear from its definition as a homology module, (St_{G})_{U} ≅ (St_{G})^{U}. Now Smith’s theorem [117] says that (St_{G})^{U} ≅ St_{L}, which proves the first part of the lemma.

For the second part, we can assume that the filtration quotients are simple F_{p}W -modules. Since U ⊆ O_{p}(W), U acts trivially on any irreducible F_{p}W -module, by elementary representation theory. Hence the second part follows from the first together with a dimension consideration. □

The above lemma is usually used in conjunction with the following obvious observation.

Lemma 10.11. Let E be a non-toral elementary abelian p-subgroup of a compact Lie group G. Then the F_{p}-dimension of _{p}ZCE is no more than the dimension of one of the maximal non-toral elementary abelian p-subgroups of G, and E is a W (E)-submodule of _{p}ZCE. □

The last lemma we shall need is a concrete calculation.

Lemma 10.12. Let E be a rank 4 elementary abelian 3-group, and let W = SL_{3}(F_{3}) × 1 ⊆ GL(E). Then Hom_{W} (St(E), E) = 0.

Proof. This most easily checked by computer, e.g., using magma [11], but is indeed a sufficiently small calculation so that the computer’s algorithm with a bit of effort can be redone by hand. Alternatively one can use some ad hoc Lie theoretic arguments. (We are grateful to A. Kleschev and H. H. Andersen for sketching a couple of such arguments to us— however, since these arguments are rather involved compared to the size of the calculation at hand we will not provide them here.) □

Before we start going through the exceptional groups, we need to introduce a bit of notation. For an F_{p}-vector space E = ⟨e_{1}, …, e_{n}⟩,we let E_{ij...} denote the subspace generated by e_{i}, e_{j}, . . .. Likewise we let P_{ij...} (resp. U_{ij...}) denote the parabolic subgroup (resp. unipotent radical) of GL(E) corresponding to the simple roots α_{i}, α_{j}, … in the standard basis and notation. For example in GL_{3}(F_{p}), U_{2} is the subgroup

[ \begin{matrix}1 & ∗ & ∗ \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ] .

Proof of Theorem 10.1 when X = E_{8}_{5}^{∧}, F_{4}_{3}^{∧}, E_{7}_{3}^{∧}, E_{6}_{3}^{∧}, or E_{8}_{3}^{∧}. By Lemma 10.8 it is enough to see that Hom_{W (E)}(St(E),_{p}(Z(C_{G}(V))) = 0. We now show this for all the E in the list. (E_{8}, 5): By [63, Lem. 10.3] E_{8} has, up to conjugacy, one non-toral elementary abelian 5-subgroup E, which has rank 3, Weyl group SL_{3}(F_{5}), and (since E is necessarily maximal) E = _{p}ZC_{G}E. Since St(E) is an irreducible module of dimension 125 we have that Hom_{W} (St(E), E) = 0. (F_{4}, 3): By [63, Thm. 7.4] there is, up to conjugacy, one non-toral elementary abelian 3group E, which has rank 3, Weyl group W = SL(E), and E = _{p}ZC_{G}(E). Since St(E) is an irreducible SL(E)-module of dimension 27, Hom_{W} (St(E),_{p}ZCE) = 0. (E_{7}, 3): By Theorem 8.18 E_{7} has, up to conjugacy, two non-toral elementary abelian 3subgroups E_{2E_{7}}^{3} and E_{2E_{7}}^{4} of rank 3 and 4 respectively. Since W (E_{2E_{7}}^{3}) = SL_{3}(F_{3}) a dimension consideration as above gives Hom_{W (E_{2E_{7}}^{3})}(St(E_{2E_{7}}^{3}),_{p}ZC_{G}E_{2E_{7}}^{3}) = 0. For E_{2E_{7}}^{4} (whose Weyl group is listed in Theorem 8.18) we use Lemma 10.10(2), taking U = U_{23}, which immediately gives that also Hom_{W (E_{2E_{7}}^{4})}(St(E_{2E_{7}}^{4}),_{p}ZC_{G}E_{2E_{7}}^{4}) = 0. (E_{6}, 3): By Theorem 8.9 E_{6} has eight non-toral elementary abelian 3-groups all of rank less than or equal to four. We follow the notation of this theorem. By Lemma 10.10(2), Hom_{W} (St(E),_{p}ZC_{G}E) = 0 when E = E_{E_{6}}^{2b}, E_{E_{6}}^{3b}, E_{E_{6}}^{3c} or E_{E_{6}}^{4a} (taking U = 1, 1, U_{2}, and U_{1} respectively). For E = E_{E_{6}}^{2a}, E_{E_{6}}^{3a}, E_{E_{6}}^{3d}, and E_{E_{6}}^{4b} we use that by Theorem 8.9 E = _{p}ZC_{G}E in these cases (a fact that we did not need above), and also −1 ∈ W (E) so Lemma 10.9 applies to show that Hom_{W} (St(E),_{p}ZC_{G}E) = 0. (E_{8}, 3): By Theorem 8.15 E_{8} has seven conjugacy classes of non-toral subgroups. If E = E_{E_{8}}^{3a}, E_{E_{8}}^{3b}, E_{E_{8}}^{4b}, or E_{E_{8}}^{4c} then Lemma 10.10 shows that Hom_{W} (St(E),_{p}ZC(E)) = 0 (taking U = U_{1}, 1, U_{2}, and U_{23} respectively). (Note that we do not need to know _{p}ZC_{G}(E) exactly since the rough bound from Lemma 10.11 will do.) We now consider E = E_{E_{8}}^{4a}. By Theorem 8.15 we have that _{p}Z(C_{G}(E)) = E, and by Lemma 10.12 Hom_{W (E)}(St(E), E) = 0. Suppose that E = E_{E_{8}}^{5a}. Then E has an invariant subspace E_{1} upon which U = U_{234} acts trivially. Now Hom_{W} (St(E), E_{1}) = Hom_{W/U}(St(E_{1}) ⊗ St(E_{2345}), E_{1}) = 0, where we use Lemma 10.10 (to see that St(E)^{U} ≅ St(E_{1}) ⊗ St(E_{2345})) and Lemma 10.9 (using that ε_{1} act trivially on St(E_{1}) but fixed point free on E_{1}). Now Hom_{W} (St(E), E/E_{1}) = Hom_{W/U}(St(E_{1}) ⊗ St(E_{2345}), E/E_{1}) = 0 by Lemma 10.12. Suppose that E = E_{E_{8}}^{5b}. Take U = U_{123} and note that E_{1234} is an invariant subspace under W. Then Hom_{W} (St(E), E_{1234}) = Hom_{W/U}(St(E_{1234}), E_{1234}) = 0 since −1 ∈ Sp(E_{1234}) ⊆ W/U. By [4] (or a direct calculation)

Hom_{Sp(E_{1234})}(St(E_{1234}), E/E_{1234}) = 0,

which shows Hom_{W} (St(E), E) = 0 as well. This exhausts the list. □

10.3. The non-toral part for the projective unitary groups. We now embark in proving Theorem 10.1 for X = P U(n)_{p}^{∧}. We will throughout this subsection use the notation for elementary abelian p-subgroups of X introduced in Section 9.

We first state the toral case.

Lemma 10.13. Let X = P U(n)_{p}^{∧}. Then

lim_{A(X)}^{i}F_{j}^{tor} = \{ \begin{matrix}Z/2 & if n = p = 2, i = 0 and j = 1 \\ 0 & otherwise. \end{matrix}

Proof. If n ≠ 2 then it is immediate to check that for an arbitrary reflection s ∈ W_{X} then T̆^{⟨s⟩} is connected so σ(s) = T̆^{⟨s⟩} by definition of σ(s). Hence if n ≠ 2 or p odd the lemma follows by Lemma 10.5.

Now suppose that X = P U(2)_{2}^{∧}. Since for the non-trivial V ⊆ T̆ we have that W_{X}(V)_{1} is trivial and C_{X}(V)_{1} ≅ T the first part of Lemma 10.5 still apply to finish the proof also in this case. □

We next record the following general lemma, which is obvious from the Künneth formula.

Lemma 10.14. Suppose D_{1} and D_{2} are two categories with only finitely many morphisms. Let CD_{i} be “the cone on D_{i}” i.e., the category constructed from D_{i} by adding an initial object e to D_{i}, and let D_{1} ⋆ D_{2} = CD_{1} × CD_{2} \ (e, e), “the join of D_{1} and D_{2}” (see [109, §1]). If F_{i} : CD_{i} → Z_{p}- mod, i = 1, 2 are functors then

C^{∗}(CD_{1} × CD_{2}, D_{1} ⋆ D_{2}; F_{1} ⊗ F_{2}) ≅ C^{∗}(CD_{1}, D_{1}; F_{1}) ⊗ C^{∗}(CD_{2}; D_{2}; F_{2}).

In particular if one of the chain complexes has torsion free homology or if everything is defined over F_{p} then

H^{∗}(CD_{1} × CD_{2}, D_{1} ⋆ D_{2}; F_{1} ⊗ F_{2}) ≅ H^{∗}(CD_{1}, D_{1}; F_{1}) ⊗ H^{∗}(CD_{2}; D_{2}; F_{2}).
□

The following result gives that certain filtration quotients have (almost) vanishing cohomology.

Theorem 10.15. Set X = P U(n)_{p}^{∧} and fix an integer r > 0. Let F^{r} : A(X) → Z_{p}- mod denote the functor on objects given by F_{j}^{r}(E) = π_{j}(BZC_{X}(E)) if E is of the form Γ̄_{r} × Ā (in the notation of Section 9) and zero otherwise. Writing n = p^{r}k we have that

lim_{A(X)}^{i}F_{j}^{r} = \{ \begin{matrix}Z/2 & if j = i = k = r = 1 and p = 2 \\ 0 & otherwise. \end{matrix}

Proof. Define a functor F̃_{j}^{r} by F̃_{j}(E) = π_{j}(BZC_{P U(k)}(Ā)_{p}^{∧}) if E = Γ̄_{r} × Ā for a fixed r and zero otherwise. This is a subfunctor of F_{j}^{r} via the identification P U(k) ≅ C_{P U(n)}(Γ̄_{r})_{1}. Set

˜F̃_{j}^{r} = F_{j}^{r}/F̃_{j}^{r} and observe that this is the trivial functor unless j = 1 where it is given by

˜F̃_{j}^{r}(E) = Γ̄_{r} if E is of the form Γ̄_{r} × Ā and zero otherwise.

Consider the category

D = A^{e}(X)_{⊆Γ̄_{r}} × A^{e}(P U(k)_{p}^{∧}) \ (e, e)

where the superscript e means that we do not exclude the trivial subgroup. We have a natural inclusion of categories ι : D → A(X) on objects given by (V, Ā) ↦ V × Ā.

Step 1: We claim that this map induces an equivalence

lim_{A(X)}^{∗}F_{j}^{r} → lim_{D}^{∗}F_{j}^{r}.

By filtering the functor and using Nakayama’s lemma it is enough to show this for F̃_{j}^{r} ⊗ F_{p}

and ˜ F̃_{j}^{r} ⊗ F_{p}. We can furthermore replace these functors by the subquotient functors which are only concentrated on one subgroup Γ̄_{r} × Ā.

Consider first such a subquotient of F̃_{j}^{r} ⊗ F_{p}. In this case the formula of Oliver [103, Prop. 4] together with Lemma 10.14 shows that the higher limits on both sides are only non-zero in a single degree, where the map identifies with the restriction map

Hom_{W_{X}(Γ̄_{r}×Ā)}(St(Γ̄_{r} × Ā), π_{j}(BZC_{P U(k)}(Ā)) ⊗ F_{p}) →
Hom_{Sp(Γ̄_{r})×W_{P U(k)}(Ā)}(St(Γ̄_{r}) ⊗ St(Ā), π_{j}(BZC_{P U(k)}(Ā)) ⊗ F_{p}).

Now note that the elements U in W_{P U(n)}(Γ̄_{r} × Ā) which sends Γ̄_{r} to Ā act trivially on the target by Theorem 9.1. Furthermore we have by the theorem of Steve Smith [117] (and self-duality of the Steinberg module) that St(Γ̄_{r} × Ā)_{U} ≅ St(Γ̄_{r} × Ā)^{U} ≅ St(Γ̄_{r}) ⊗ St(Ā), where the (−)_{U} and (−)^{U} denotes coinvariants and invariants respectively. Hence this map is an isomorphism. The case of a subquotient of ˜ F̃_{j}^{r} ⊗ F_{p} is completely analogous. This shows the isomorphism.

Step 2: We now proceed to calculate the higher limits over D, which we do by calculating the limits of F̃_{j}^{r} and ˜ F̃_{j}^{r}. We first consider ˜ F̃_{j}^{r}. We have already remarked that only

˜F̃_{1}^{r} ≠ 0. Furthermore if k ≠ 1 then H^{∗}(CA^{tor}(P U(k)), A^{tor}(P U(k)); Z_{p}) = 0 by [47, §8] so lim_{D}^{∗} ˜ F̃_{j}^{r} = 0 by Lemma 10.14. If k = 1 we get lim_{D}^{i} ˜ F̃^{r} ≅ Hom_{Sp(Γ̄_{r})}(St(Γ̄_{r}), Γ̄_{r}) if i = 2r−1 and zero otherwise, by [103, Prop. 4].

Now consider F̃_{j}^{r}. By Lemma 10.14

lim_{D}^{i}F̃_{j}^{r} = Hom_{Sp(Γ̄_{r})}(St(Γ̄_{r}), Z_{p}) ⊗ lim_{Ā∈A^{tor}(P U(k)_{p}^{∧})}^{i−2r}π_{j}(BZC_{P U(k)_{p}^{∧}}(Ā)).

By Lemma 10.13 lim_{A^{tor}(P U(k)_{p}^{∧})}^{i−2r} π_{j}(BZC_{P U(k)_{p}^{∧}}(Ā)) = 0 unless p = k = 2, j = 1 and

i − 2r = 1 where we get lim_{A^{tor}(P U(2)_{2}^{∧})}^{1} π_{j}(BZC_{P U(2)_{2}^{∧}}(Ā)) = Z/2.

By an argument of H. H. Andersen and C. Stroppel [4] we have that

Hom_{Sp(Γ̄_{r})}(St(Γ̄_{r}), Z_{p}) = 0

for all r and p.

To sum up we get that

lim_{A(P U(n))}^{i}F_{j}^{r} ≅ lim_{D}^{i} ˜F̃_{j}^{r} ≅ Hom_{Sp(Γ̄_{r})}(St(Γ̄_{r}), Γ̄_{r})

if j = k = 1 and i = 2r − 1 and zero otherwise.

However the same argument of H. H. Andersen and C. Stroppel [4] shows that

Hom_{Sp(Γ̄_{r})}(St(Γ̄_{r}), Γ̄_{r}) = 0

unless r = 1 and p = 2 where it equals F_{2}. (Note that this is obvious if p is odd by Lemma 10.9.) This shows the wanted formula. □

Remark 10.16. Note that slightly non-trivial statements from [4] are only used above for p = 2 and furthermore become trivial when r = 1, where Sp(Γ̄_{1}) ≅ SL(Γ̄_{1}), and this is in fact the only case which involve obstruction groups in the range needed for the proof of (the p = 2 version of) Theorem 1.4.

Remark 10.17. It is in fact possible to give a short proof of Smith’s theorem in the special case used above, using the geometric definition of the Steinberg module St(E) via flags.

Proof of Theorem 10.1 for X = P U(n)_{p}^{∧}. Lemma 10.15 and 10.13 directly shows the conclusion unless n = 2 and p = 2, so assume we are in this case. By writing down the definition it follows that lim_{A(X)}^{0} F_{1} = 0. Now, consider the exact sequence of functors 0 → F_{1}^{1} → F_{1} → F_{1}^{tor} → 0. The long exact sequence of higher limits starts out as

0 → 0 → 0 → lim_{A(X)}^{0}F_{1}^{tor} → lim_{A(X)}^{1}F_{1}^{1} → lim_{A(X)}^{1}F_{1} → 0 → ⋯

So, since lim_{A(X)} ^{0}F_{1}^{tor} ≅ Z/2 ≅ lim_{A(X)} ^{1}F_{1}^{1} we get that lim_{A(X)}^{i} F_{1} = 0 for i > 0 as well. This concludes the proof of this last case of Theorem 10.1. □

11. Appendix: The classification of finite Z_{p}-reflection groups

The purpose of this appendix is to give a classification of finite Z_{p}-reflection groups extending and simplifying work of Notbohm [95, 96] who gave a classification for odd primes p.

We start by recalling some definitions. Let R be an integral domain with field of fractions K. An R-reflection group is a pair (W, L) where L is a finitely generated free R-module, and W is a subgroup of Aut(L) generated by elements α such that 1 − α has rank one viewed as a matrix over K. Two finite R-reflection groups (W, L) and (W′, L′) are called equivalent, if we can find a R-linear isomorphism φ : L → L′ such that the group φW φ^{−1} equals W′. A finite R-reflection group (W, L) is said to be irreducible if the corresponding representation of W on L ⊗_{R} K is irreducible. If R has characteristic zero we define the character field of an R-reflection group (W, L) as the field extension of Q generated by the values of the character of the representation W ↪ Aut(L). For R = Z_{p} or Q_{p} we define an exotic R-reflection group to be a finite irreducible R-reflection group with character field larger than Q.

The classification of finite Z_{p}-reflection groups is based on the work of Clark and Ewing [29], which is again based on the classification of finite C-reflection groups by Shephard and Todd [115]. Clark and Ewing showed that there is a bijection between finite Q_{p}-reflection groups and finite C-reflection groups whose character field may be embedded in Q_{p}. The classification of finite complex reflection groups [115] is as follows: The irreducible ones fall into 3 infinite families (in the following called family 1, 2 and 3) and 34 sporadic cases (in the following labeled G_{i}, 4 ≤ i ≤ 37). Moreover any finite complex reflection group can be written as a direct product of irreducible finite complex reflection groups, cf. [55, Rem. 2.3] (in fact this holds over any field of characteristic 0).

It is convenient to split family 2 further depending on the character field. The associated complex reflection group is the group G(m, r, n) (where m, r and n are integers with m, n ≥ 2, r ≥ 1, r | m and (m, r, n) ≠ (2, 2, 2)) from [115, p. 277] which consists of monomial n×nmatrices such that the non-zero entries are m’th roots of unity and the product of the non-zero entries is an (m/r)’th root of unity. Thus G(m, r, n) is the semidirect product of its subgroup A(m, r, n) of diagonal matrices with the subgroup of permutation matrices. Let ζ_{m} = e^{2πi/m}. For n ≥ 3 or n = 2 and r ≠ m the character field of G(m, r, n) equals Q(ζ_{m}), and for n = 2 and r = m it equals Q(ζ_{m} + ζ_{m}^{−1}) (see [29, p. 432–433]). We label the two cases family 2a and 2b respectively.

A complete list of the irreducible finite complex reflection groups, their character fields and the primes for which these embed in Q_{p} can be found in [77, p. 165] or [5, Table 1].

If (W, V) is a finite Q_{p}-reflection group, then by [34, Prop. 23.16] we can find a finitely generated Z_{p}W -submodule L ⊆ V with L ⊗ Q = V. Thus any finite Q_{p}-reflection may be obtained from a finite Z_{p}-reflection group by extension of scalars, but this may give several inequivalent Z_{p}-reflection groups. The following result extends [96, Thm. 1.5 and Prop. 1.6] to all primes.

Theorem 11.1 (The classification of finite Z_{p}-reflection groups). Let (W, L) be a finite Z_{p}-reflection group. Then there exists a decomposition

(W, L) = (W_{1} × W_{2}, L_{1} ⊕ L_{2})

where (W_{1}, L_{1}) ≅ (W_{G}, L_{G} ⊗ Z_{p}), for some (non-unique) compact connected Lie group G with integral lattice L_{G}, and (W_{2}, L_{2}) is a (up to permutation unique) direct product of exotic Z_{p}-reflection groups.

The exotic Z_{p}-reflection groups are in natural one-to-one correspondence with the exotic Q_{p}-reflection groups, i.e. the finite irreducible Q_{p}-reflection groups on Clark-Ewing’s list whose character field is not contained in Q.

If (W, L) is any exotic Z_{p}-reflection group, then L ⊗ F_{p} is an irreducible F_{p}W -module, and in particular we have (L ⊗ Z/p^{∞})^{W} = 0 and H_{0}(W; L) = 0.

Remark 11.2. For odd primes p the last two results says by definition that any exotic Z_{p}-reflection group is respectively center-free and simply connected, cf. [95].

Note also that [96, Thm. 1.5] imposes the unnecessarily strong condition that the invariant ring Z_{p}[L]^{W} is a polynomial algebra, but this condition is not actually used in [96].

In the proof we shall use the following result concerning the mod-p reduction GL_{n}(Z_{p}) → GL_{n}(F_{p}).

Lemma 11.3. Let G ⊆ GL_{n}(Z_{p}) be a finite subgroup. Then the composition G ↪ GL_{n}(Z_{p}) → GL_{n}(F_{p}) of the inclusion with the mod-p reduction is injective if p is odd. For p = 2 the kernel of the composition is an elementary abelian 2-subgroup of rank at most n^{2}. In particular the kernel is contained in O_{2}(G), the largest normal 2-subgroup of G.

Proof. For p odd this follows from [116, Pf. of Lem. 10.7.1]. For p = 2 the same argument shows that the composition G ↪ GL_{n}(Z_{2}) → GL_{n}(Z/4) is injective. As the kernel of GL_{n}(Z/4) → GL_{n}(F_{2}) is an elementary abelian 2-group of rank n^{2} the result follows. □

Proof of Theorem 11.1. We start by showing that for any exotic Q_{p}-reflection group (W, V) we can find a finitely generated Z_{p}W -submodule L ⊆ V with L ⊗ Q = V, such that L ⊗ F_{p} is an irreducible F_{p}W -module.

Assume first that p ∤ |W |. By [34, Prop. 23.16] we can find a finitely generated Z_{p}W - submodule L ⊆ V with L ⊗ Q = V. It follows from [33, 75.6 and 76.15] that L ⊗ F_{p} is automatically an irreducible F_{p}W -module.

Assume now that W has order divisible by p. From Clark-Ewing’s list we see that the only exotic Q_{p}-reflection groups satisfying this condition are the groups G(m, r, n) from family 2a or one of the groups G_{12} for p = 3, G_{24} for p = 2, G_{29} and G_{31} for p = 5 or G_{34} for p = 7.

In case W = G(m, r, n) from family 2a we get the extra conditions m ≥ 3, p ≡ 1 (mod m) and p ≤ n. Note in particular that n ≥ 3. The description above directly gives a representation with entries in Z_{p} since the multiplicative group of Z_{p} contains the (p − 1)’st roots of unity. Let L = (Z_{p})^{n} be the natural Z_{p}W -module, i.e. the set of columns with entries in Z_{p}. Assume that 0 ≠ M ⊆ L ⊗ F_{p} is a F_{p}W -submodule of L ⊗ F_{p}. Choose x ∈ M with x ≠ 0 and let θ ∈ F_{p} be a primitive m’th root of unity. Since W contains the permutation matrices and the diagonal matrix diag(θ, θ^{−1}, 1, …, 1) we see that M contains an element of the form x′ = (x_{1}, x_{2}, 0, …, 0)^{T} with x_{1} ≠ 0. Since n ≥ 3, W also contains the diagonal matrix diag(θ, 1, θ^{−1}, 1, …, 1) and hence M contains ((1 − θ)x_{1}, 0, …, 0)^{T}. As θ ≠ 1 and W contains all permutation matrices we conclude that M = L ⊗ F_{p}, proving the claim for the groups from family 2a.

Next consider W = G_{12} at p = 3. Since W is isomorphic to GL_{2}(F_{3}), Lemma 11.3 shows that for any finitely generated Z_{3}W -submodule L ⊆ (Q_{3})^{2} of rank 2, we may identify L⊗F_{3} as the natural F_{3}W -module. In particular L ⊗ F_{3} is an irreducible F_{3}W -module.

For W = G_{24} at p = 2 we have W ≅ Z/2 × GL_{3}(F_{2}). Hence Lemma 11.3 shows that for any finitely generated Z_{2}W -submodule L ⊆ (Q_{2})^{3} of rank 3, we may identify L ⊗ F_{2} as the F_{2}(Z/2 × GL_{3}(F_{2}))-module where Z/2 acts trivially and GL_{3}(F_{2}) acts naturally. In particular L ⊗ F_{2} is an irreducible F_{2}W -module.

Next consider the groups G_{29} and G_{31} at p = 5. Since G_{29} is contained in G_{31} it suffices to show the result for W = G_{29}. The representation in [115, p. 298] is defined over Z[\frac{1}{2}, i] and hence we get a representation over Z_{5} by mapping i to a primitive 4’th root of unity in Z_{5}. Let L = (Z_{5})^{4} be the natural Z_{5}W -module. There are 40 reflections in G_{29}: The 24 reflections in the hyperplanes of the form x_{j} − i^{α}x_{k} = 0, j ≠ k and the 16 reflections in the hyperplanes of the form ∑_{j=1}^{4} i^{α_{j}}x_{j} = 0 with ∑_{j=1}^{4} α_{j} ≡ 0 (mod 4). In particular G_{29} contains the reflections in the hyperplanes x_{j} −x_{k} = 0 and thus G_{29} contains all permutation matrices. The product of the reflections in the hyperplanes x_{1} − ix_{2} = 0 and x_{1} − x_{2} = 0 equals the diagonal matrix diag(i, −i, 1, 1) and thus this element is also contained in G_{29}. Now the same argument used in the case of the groups from family 2a shows that L ⊗ F_{5} is an irreducible F_{5}W -module.

The argument for the group W = G_{34} at p = 7 is similar. The representation given in [115, p. 298] is defined over Z[\frac{1}{3}, ω], ω = ζ_{3} and hence we get a representation over Z_{7} by mapping ω to a primitive 3’rd root of unity in Z_{7}. Let L = (Z_{7})^{6} be the natural Z_{7}W - module. There are 126 reflections in G_{34}: The 45 reflections in the hyperplanes of the form x_{j} − ω^{α}x_{k} = 0, j ≠ k and the 81 reflections in the hyperplanes of the form ∑_{j=1}^{6} ω^{α_{j}}x_{j} = 0 with ∑_{j=1}^{6} α_{j} ≡ 0 (mod 3). In particular G_{34} contains all permutation matrices. The product of the reflections in the hyperplanes x_{1} − ωx_{2} = 0 and x_{1} − x_{2} = 0 equals the diagonal matrix diag(ω, ω^{2}, 1, 1, 1, 1) and thus this element is also contained in G_{34}. As above we then see that L ⊗ F_{7} is an irreducible F_{7}W -module.

This proves the claim made above that for any exotic Q_{p}-reflection group (W, V) we can find a finitely generated Z_{p}W -submodule L ⊆ V with L ⊗ Q = V, such that L ⊗ F_{p} is an irreducible F_{p}W -module. It now follows from [114, Ex. 15.3] that, up to scaling, L is the only such Z_{p}W -submodule. This gives the natural bijection between exotic Z_{p}-reflection groups and exotic Q_{p}-reflection groups.

Since L ⊗ F_{p} is an irreducible F_{p}W -module we also conclude that (L ⊗ F_{p})^{W} = 0 and H_{0}(W; L⊗F_{p}) = 0 since the W -action on L⊗F_{p} is non-trivial. Hence we get (L⊗Z/p^{∞})^{W} = 0 as claimed. We also see that multiplication by p is surjective on H_{0}(W; L) and from this we obtain H_{0}(W; L) = 0 by Nakayama’s lemma. This proves the part of the theorem pertaining to exotic Z_{p}-reflection groups.

Now consider a finite Z_{p}-reflection group (W, L) such that we have a direct sum decomposition L ⊗ Q = V_{1} ⊕ V_{2} as Q_{p}W -modules. Let W_{1} (resp. W_{2}) be the subgroup of W which fixes V_{2} (resp. V_{1}) pointwise. It it easy to see (cf. [50, Prop. 6.3]) that (W_{i}, V_{i}) is a Q_{p}-reflection group and that we get the decomposition (W, L ⊗ Q) = (W_{1} × W_{2}, V_{1} ⊕ V_{2}).

We now claim that if (W_{2}, V_{2}) is an exotic Q_{p}-reflection group, then we have the decomposition (W, L) = (W_{1} × W_{2}, L_{1} ⊕ L_{2}) with L_{i} = L ∩ V_{i}. Let α : L_{1} ⊕ L_{2} ⟶ L be the addition map. As in [50, Pf. of Thm. 1.5] it suffices to prove that α ⊗ Z/p^{∞} : (L_{1} ⊗ Z/p^{∞}) ⊕ (L_{2} ⊗ Z/p^{∞}) ⟶ L ⊗ Z/p^{∞} is injective. Assume that (x_{1}, x_{2}) is in the kernel of α ⊗ Z/p^{∞}, x_{i} ∈ L_{i} ⊗ Z/p^{∞}. Thus x_{1} + x_{2} = 0. If s ∈ W_{2} is a reflection we have by definition s · x_{1} = x_{1} and hence s also fixes x_{2} = −x_{1}. Since W_{2} is generated by reflections we get x_{2} ∈ (L_{2} ⊗ Z/p^{∞})^{W} and hence x_{2} = 0 by the results already proved for exotic Z_{p}-reflection groups. Hence x_{1} = 0 as well, and thus α ⊗ Z/p^{∞} is injective proving the claim.

Since any finite Q_{p}-reflection group may be decomposed into a (up to permutation unique) product of finite irreducible ones, we see by using the claim repeatedly that any finite Z_{p}- reflection group (W, L) may be decomposed as a product (W, L) ≅ (W_{1} × W_{2}, L_{1} ⊕ L_{2}) where (W_{1}, L_{1}) is a Z_{p}-reflection group with character field equal to Q and (W_{2}, L_{2}) is as in the theorem.

To finish the proof we thus need to show that for any finite Z_{p}-reflection group (W, L) with character field equal to Q we may find a compact Lie group G such that (W, L) is equivalent to (W_{G}, L_{G}⊗Z_{p}). We start by reducing the problem to finite Z-reflection groups. The representation W → GL(L⊗Q) is a reflection representation and hence has Schur index 1 by [29, Cor. p. 429]. Thus this representation is equivalent to a representation defined over Q. Hence [34, Cor. 30.10] applied to R = Z_{(p)} shows that there exist a (unique) finitely generated Z_{(p)}W -submodule L′ ⊆ L with L′ ⊗ Z_{p} = L. Now [34, Cor. 23.14] applied to R = Z shows that L′ contains a (non-unique) finitely generated ZW -submodule L′′ with L′ = L′′ ⊗ Z_{(p)}. We conclude in particular that (W, L) ≅ (W, L′′ ⊗ Z_{p}).

We finish the proof by showing that there exists a (non-unique) compact connected Lie group G whose Weyl group (W_{G}, L_{G}) is equivalent to (W, L′′). For each reflection s ∈ W the group {x ∈ L′′ | s(x) = −x} is an infinite cyclic group with two generators which we label ±α_{s}. Let Φ = {±α_{s} | s is a reflection in W } and L′′_{0} = (L′′)^{W}. It then follows (cf. [106, p. 85]) that (L′′, L′′_{0}, Φ) is a reduced root diagram whose associated Z-reflection group equals (W, L′′) (see [15] for definitions). From the classification of compact connected Lie groups ([15, §4,no. 9, Prop. 16]) it then follows that there exists a compact connected Lie group G whose root diagram equals (L′′, L′′_{0}, Φ). In particular (W_{G}, L_{G}) is equivalent to (W, L′′) and we are done. □

The following result answers the question of when two connected compact Lie groups give rise to the same p-compact group. For a compact connected Lie group G, let H be the direct product of Z(G)_{1} with the universal cover of the derived group of G. We have a canonical finite covering homomorphism φ : H → G with finite kernel (cf. [15, Prop. 1.4.4]). If p is a prime number, we let Cov^{p′}(G) denote the covering of G corresponding to subgroup of π_{1}(G) given as the preimage of the Sylow p-subgroup of π_{1}(G)/φ(π_{1}(H)).

Theorem 11.4 (Addendum to Theorem 1.1 and 11.1). Let G and G′ be two compact connected Lie groups and p a prime number. Then

(1) (W_{G}, L_{G}) and (W_{G′}, L_{G′}) are equivalent if and only if G is isomorphic to G′ up to the substitution of direct factors isomorphic to Sp(n) with SO(2n + 1).

(2) (W_{G}, L_{G} ⊗ Z_{2}) and (W_{G′}, L_{G′} ⊗ Z_{2}) are equivalent if and only if Cov^{2′}(G) and Cov^{2′}(G′) are isomorphic up to the substitution of direct factors isomorphic to Sp(n) with SO(2n + 1). Moreover the following conditions are equivalent:

(a) (W_{G}, L_{G} ⊗ Z_{2}, L_{G⟨1⟩} ⊗ Z_{2}) and (W_{G′}, L_{G′} ⊗ Z_{2}, L_{G′⟨1⟩} ⊗ Z_{2}) are equivalent.

(b) Cov^{2′}(G) is isomorphic to Cov^{2′}(G′).

(c) (BG)_{2}^{∧} ≃ (BG′)_{2}^{∧}.

(3) For p odd the following conditions are equivalent:

(a) (W_{G}, L_{G} ⊗ Z_{p}) and (W_{G′}, L_{G′} ⊗ Z_{p}) are equivalent.

(b) Cov^{p′}(G) and Cov^{p′}(G′) are isomorphic up to the substitution of direct factors isomorphic to Sp(n) with Spin(2n + 1).

(c) (BG)_{p}^{∧} ≃ (BG′)_{p}^{∧}.

Sketch of proof: By [106, §4] or [75, Prop. 3.2(vi)] we can recover the root datum of a compact connected Lie group from its integral lattice up to substitution of direct factors isomorphic to Sp(n) with direct factors isomorphic to SO(2n + 1). Part (1) now follows.

Now, suppose that G is a compact connected Lie group of the form H/K where H is a direct product of a torus with a simply connected compact Lie group and K is a finite central p-group. Suppose moreover that G does not contain any direct factors isomorphic to Sp(n). By Proposition 7.4 the fundamental group of G equals the coinvariants (L_{G})_{W} and hence L_{H} = (L_{G})^{W} ⊕SL_{G}. This shows that (W, L_{H} ⊗Z_{p}) can be reconstructed from (W, L_{G}⊗Z_{p}). By the classification of simply connected compact Lie groups we can for p = 2 reconstruct H from (W, L_{H} ⊗Z_{p}). For p odd the only ambiguity arises from direct factors isomorphic to Sp(n) or Spin(2n+1). However, if p is odd then by the assumption on G, H cannot contain any direct factors isomorphic to Sp(n), and we conclude that in all cases we can reconstruct H from (W, L_{G} ⊗ Z_{p}). Since K is the cokernel of the inclusion L_{H} ⊗ Z_{p} → L_{G} ⊗ Z_{p}, we can also recover the inclusion K ⊆ L_{H} ⊗ Z/p^{∞} = Z̆(H) ⊆ Z(H) using [55, Thm. 1.4] for the middle equality. This shows that we can recover G from (W, L_{G} ⊗ Z_{p}).

The above analysis directly shows the first claim in (2) as well as (3a) ⇔ (3b). From the first claim in (2), (2a) ⇔ (2b) follows, since Sp(n) and SO(2n+1) have different Z_{2}-reflection data (W_{G}, L_{G} ⊗ Z_{2}, L_{G⟨1⟩} ⊗ Z_{2}). The implications (2b) ⇒ (2c) ⇒ (2a) are clear.

To finish off we remark that (3c) ⇒ (3a) is clear and (3b) ⇒ (3c) follows since BSO(2n+ 1)_{p}^{∧} is homotopy equivalent to BSp(n)_{p}^{∧} either by a very special case of our main Theorem 1.1 or by the original proof due to Friedlander [59]. □

12. Appendix: Invariant rings of finite Z_{p}-reflection groups, p odd (following Notbohm)

The purpose of this appendix is to recall Notbohm’s determination [96] of finite Z_{p}- reflection groups (W, L), p odd, such that invariant ring Z_{p}[L]^{W} is a polynomial algebra.

Before stating it let us however for easy reference recall the following ‘classical’ characterizations of being a ‘p-torsion free’ p-compact group, which has a proof by non-case-by-case arguments which we will sketch.

Theorem 12.1. Let X be a connected p-compact group with maximal torus normalizer T. The following statements are equivalent:

(1) H^{∗}(BX; Z_{p}) is torsion free.

(2) H^{∗}(X; Z_{p}) is torsion free.

(3) H^{∗}(X; Z_{p}) is an exterior algebra over Z_{p} with generators in odd degrees (or equivalently with F_{p} instead of Z_{p}).

(4) H^{∗}(BX; Z_{p}) is a polynomial algebra over Z_{p} with generators in even degree (or equivalently with F_{p} instead of Z_{p}).

(5) H^{∗}(BX; Z_{p}) →^{≅} H^{∗}(BT; Z_{p})^{W_{X}}.

We now give Notbohm’s classification. The first part (which is a general argument reducing to the simply connected case) is [96, Thm. 1.3] and the second (which is a case-by-case argument of the simply connected cases) is a slight extension of [96, Thm. 1.4]. For the benefit of the reader we give a streamlined proof of the second part. Recall that for a finite Z_{p}-reflection group (W, L) we define SL to be the submodule of L generated by elements of the form (1 − w)x with w ∈ W and x ∈ L. We call (W, L) simply connected if L = SL′ for some Z_{p}W -lattice L′ (note that for p odd this is equivalent to SL = L since S^{2}L′ = SL′, cf. the discussion of Z_{p}-reflection data in the introduction).

Theorem 12.2 (Finite Z_{p}-reflection groups with polynomial invariants, p odd). Let p be an odd prime and (W, L) a finite Z_{p}-reflection group. Then we have the following statements:

(1) Z_{p}[L]^{W} is a polynomial algebra if and only if Z_{p}[SL]^{W} is a polynomial algebra and the group of coinvariants L_{W} is torsion free.

(2) Suppose (W, L) is simply connected. The following conditions are equivalent: (a) Z_{p}[L]^{W} is a polynomial algebra.

(b) F_{p}[L ⊗ F_{p}]^{W} is a polynomial algebra.

(c) (W, L) is not equivalent to (W_{G}, L_{G} ⊗Z_{p}) for the following pairs (G, p): (F_{4}, 3), (3E_{6}, 3), (2E_{7}, 3), (E_{8}, 3) and (E_{8}, 5).

In particular, if X is an exotic p-compact group then Z_{p}[L_{X}]^{W_{X}} is a polynomial algebra and if (W, L) = (W_{G}, L_{G} ⊗ Z_{p}) for a connected compact Lie group G then Z_{p}[L_{G} ⊗ Z_{p}]^{W_{G}} is a polynomial algebra if and only if H^{∗}(G; Z_{p}) is torsion free.

Sketch of proof of Theorem 12.1. The equivalence of (1), (2), (3), and (4) are old H-space and loop space arguments which we first very briefly sketch. By a Bockstein spectral sequence argument (cf. e.g., [77, §11-2]) H^{∗}(X; Z_{p}) torsion free if and only if H^{∗}(X; Z_{p}) is an exterior algebra on odd dimensional generators so (2) is equivalent to (3). This is again equivalent to that H^{∗}(BX; Z_{p}) is a polynomial algebra on even dimensional generators (using the Eilenberg-Moore and the cobar spectral sequence; see e.g., [77, §7-4]), so (3) is equivalent to (4). Obviously (4) implies (1). Furthermore (1) implies (2), since if we had a torsion class in H^{∗}(X; Z_{p}) of lowest degree then it would correspond to a torsion class in H^{∗}(BX; Z_{p}) by the path-loop fibration. Also (4) implies (1) since if H^{∗}(BX; Z_{p}) is concentrated in even degrees then it is torsion free by the universal coefficient theorem.

That (5) implies (1) is obvious since (5) implies that H^{∗}(BX; Z_{p}) is concentrated in even degrees. The fact that (1) also imply (5) requires more machinery and is probably first found in [44, Thm. 2.11] (see also [87, Thm. 4.2(1)])—we quickly sketch an argument. We want to show that the map r : H^{∗}(BX; Z_{p}) → H^{∗}(BT; Z_{p})^{W} is an isomorphism. By [48, Thm. 9.7(3)]

(12.1)  H^{∗}(BX; Z_{p}) ⊗ Q →^{≅} H^{∗}(BT; Z_{p})^{W} ⊗ Q.

This implies by comparing Krull dimensions that the number of polynomial generators equals the rank of T. Since H^{∗}(BT; F_{p}) is finitely generated over H^{∗}(BX; F_{p}) by [48, Prop. 9.11] it follows by comparing Krull dimensions again that H^{∗}(BX; F_{p}) → H^{∗}(BT; F_{p}) is injective. Hence H^{∗}(BX; Z_{p}) → H^{∗}(BT; Z_{p}) has to be injective by Nakayama’s lemma. Likewise r has to be surjective: By (12.1) the cokernel of r has to be p-torsion. Since the reduction mod p of r is still injective (as seen above) the cokernel of r has to be p-torsion free as well (since Tor(coker(r), F_{p}) = 0). □

Remark 12.3. If p is odd then F_{p}-coefficients can also be used in Theorem 12.1(5) by a Galois theory argument using Lemma 11.3. For p = 2, this is not true as can be seen by taking X = SU(2)_{2}^{∧}. See [45] for a p = 2 version.

Remark 12.4. If (W, L) is a finite Z_{p}-reflection group then Z_{p}[L]^{W} is a polynomial algebra if and only if F_{p}[L ⊗ F_{p}]^{W} is a polynomial algebra and the canonical monomorphism Z_{p}[L]^{W} ⊗ F_{p} ⟶ F_{p}[L ⊗ F_{p}]^{W} is an isomorphism as shown in [96, Lem. 2.3]. Note that this can be reformulated as saying that Z_{p}[L]^{W} is a polynomial algebra if and only if F_{p}[L⊗F_{p}]^{W} is a polynomial algebra with generators in the same degrees as the generators of Q_{p}[L⊗Q]^{W}, since dim_{Q_{p}}(Q_{p}[L ⊗ Q]^{W})_{n} = dim_{F_{p}}(Z_{p}[L]^{W} ⊗ F_{p})_{n} ≤ dim_{F_{p}}(F_{p}[L ⊗ F_{p}]^{W})_{n} for any n.

Remark 12.5. The finite Z_{3}-reflection group (W, L) = (W_{P U(3)}, L_{P U(3)} ⊗ Z_{3}) does not have invariant ring a polynomial ring (e.g., since L_{W} ≅ Z/3 is not torsion free). However a short calculation shows that F_{3}[L ⊗ F_{3}]^{W} is a polynomial ring with generators in degrees 1 and 6 (as opposed to the degrees over Q_{3} which are 2 and 3). (See also [44, Rem. 5.3].) It turns out that this example is essentially the only example since one can prove that if (W, L) is a finite Z_{p}-reflection group, p odd, such that F_{p}[L]^{W} is a polynomial algebra, then Z_{p}[L]^{W} is also a polynomial algebra unless p = 3 and (W, L) contains (W_{P U(3)}, L_{P U(3)} ⊗Z_{3}) a direct summand. We omit the proof which is an extension of the technique used in the examples in Section 7 in a preprint version of [53], which can at the time of writing be found on Wilkerson’s homepage.

Lemma 12.6. Assume that L is a finitely generated free Z_{p}-module and that W is a finite subgroup of GL(L). If p ∤ |W | and F_{p}[L ⊗ F_{p}]^{W} is a polynomial algebra, then Z_{p}[L]^{W} is also a polynomial algebra.

Proof. By assumption we have the averaging homomorphisms Z_{p}[L] ⟶ Z_{p}[L]^{W} and F_{p}[L⊗ F_{p}] ⟶ F_{p}[L ⊗ F_{p}]^{W} given by f ↦ \frac{1}{|W |} ∑_{w∈W} w · f. These are obviously surjective and hence the commutative diagram

\xymatrix{Z_{p}[L] \ar[r] \ar[d] & Z_{p}[L]^{W} \ar[d] \\ F_{p}[L ⊗ F_{p}] \ar[r] & F_{p}[L ⊗ F_{p}]^{W}}

shows that the reduction homomorphism Z_{p}[L]^{W} → F_{p}[L ⊗ F_{p}]^{W} is surjective. The result now follows easily from Nakayama’s lemma (cf. [96, Lem. 2.3]). □

Proof of Theorem 12.2. Part (1) is contained in [96, Thm. 1.3]. To prove part (2) note that by Notbohm [95] (see also [96, Thm. 1.2(iii)]), there is a unique finite irreducible simply connected Z_{p}-reflection group for each group on the Clark-Ewing list. We now go through the list, verifying the result in each case.

If p ∤ |W | the invariant ring F_{p}[L ⊗ F_{p}]^{W} is a polynomial algebra by the Shephard-Todd-Chevalley theorem ([6, Thm. 7.2.1] or [116, Thm. 7.4.1]), and thus Lemma 12.6 shows that Z_{p}[L]^{W} is a polynomial algebra.

Next, assume that (W, L) is an exotic Z_{p}-reflection group. If (W, L) belongs to family number 2 on the Clark-Ewing list, the representing matrices with respect to the standard basis are monomial and so Z_{p}[L]^{W} is a polynomial algebra by [92, Thm. 2.4].

An inspection of the Clark-Ewing list now shows that only 4 exotic cases remain, namely (G_{12}, p = 3), (G_{29}, p = 5), (G_{31}, p = 5) and (G_{34}, p = 7). In the first case we have G_{12} ≅ GL_{2}(F_{3}) and Lemma 11.3 shows that the action on L ⊗ F_{3} = (F_{3})^{2} is the canonical one. The invariant ring F_{3}[L ⊗ F_{3}]^{GL_{2}(F_{3})} was computed by Dickson [37]. In the remaining 3 cases the mod-p invariant ring was calculated by Xu [132, 133] using computer, see also Kemper and Malle [78, Prop. 6.1]. The conclusion of these computations is that in all 4 cases the invariant ring F_{p}[L ⊗ F_{p}]^{W} is a polynomial algebra with generators in the same degrees as the generators of Q_{p}[L ⊗ Q]^{W}. By Remark 12.4 we then see that Z_{p}[L]^{W} is a polynomial algebra in these cases.

The only remaining cases are the finite simply connected Z_{p}-reflection groups which are not exotic. Since p is odd and π_{1}(G) and (L_{G})_{W_{G}} only differ by an elementary abelian 2group (cf. proof of Theorem 1.7 and Remark 7.4), we may assume that (W, L) = (W_{G}, L_{G} ⊗ Z_{p}) for some simple connected Lie group G. In this case Demazure [36] shows that if p is not a torsion prime for the root system associated to G, then the invariant rings Z_{p}[L_{G} ⊗Z_{p}]^{W_{G}} and F_{p}[L_{G} ⊗ F_{p}]^{W_{G}} are polynomial algebras.

By the calculation of torsion primes for the simple root systems, [36, Prop. 8], the excluded pairs (G, p) in the last part of the theorem are exactly the cases where the root system of G has p-torsion. In these cases Kemper and Malle [78, Prop. 6.1 and pf. of Thm. 8.5] shows that F_{p}[L_{G} ⊗ F_{p}]^{W_{G}} is not a polynomial algebra. Hence in these cases Z_{p}[L_{G} ⊗ Z_{p}]^{W_{G}} is not a polynomial algebra by [96, Lem. 2.3(i)]. This proves the second claim.

Finally, let G be a compact connected Lie group with Weyl group W and integral lattice L = L_{G}. We now prove that Z_{p}[L ⊗ Z_{p}]^{W} is a polynomial algebra if and only if H^{∗}(G; Z_{p}) is torsion free. (See also [97, Prop. 1.11]). One direction follows from Theorem 12.1, so assume now that Z_{p}[L ⊗ Z_{p}]^{W} is a polynomial algebra. From Theorem 12.2 we see that Z_{p}[S(L ⊗ Z_{p})]^{W} is a polynomial algebra and that (L ⊗ Z_{p})_{W} is torsion free. Since p is odd, we have (L⊗Z_{p})_{W} = π_{1}(G)⊗Z_{p} and S(L⊗Z_{p}) = L_{G⟨1⟩} ⊗Z_{p}, cf. proof of Theorem 1.7 and Remark 7.4. From what we have proved above we conclude that H^{∗}(G⟨1⟩; Z_{p}) is torsion free. Since π_{1}(G) does not contain p-torsion, it now follows easily from the Serre spectral sequence that H^{∗}(G; Z_{p}) is torsion free. □

Remark 12.7. Let p be an odd prime and (W, L) a finite Z_{p}-reflection group. We claim that the following conditions are equivalent:

(1) Z_{p}[L]^{W} is a polynomial algebra.

(2) F_{p}[L ⊗ F_{p}]^{W} is a polynomial algebra and L_{W} is torsion free.

(3) F_{p}[SL ⊗ F_{p}]^{W} is a polynomial algebra and L_{W} is torsion free. Indeed we have (1) ⇔ (3) by Theorem 12.2 since (W, SL) can be decomposed as a direct product of irreducible finite simply connected Z_{p}-reflection groups by [95, Thm. 1.4]. The implication (1) ⇒ (2) follows from [96, Thm. 1.3 and Lem. 2.3]. Finally (2) ⇒ (3) follows from [92, Prop. 4.1] as L_{W} torsion free implies that SL ⊗ F_{p} → L ⊗ F_{p} is injective.

13. Appendix: Outer automorphisms of exotic finite Z_{p}-reflection groups

Theorem 1.1 states that the outer automorphism group of a p-compact group, p odd, equals N_{GL(L)}(W)/W, which makes it useful to have a complete case-by-case calculation of this object. Theorem 11.1 and Proposition 3.4 reduces the calculation to the case where (W, L) = (W_{G}, L_{G}⊗Z_{p}) for some connected compact Lie group G and the case where (W, L) is exotic.

The purpose of this appendix section is to provide such a calculation when (W, L) is exotic based on calculations of Broué, Malle and Michel [21, Prop. 3.13] over the complex numbers. (Information about the, perhaps more familiar, Lie case can be obtained similarly, or by a very close reading of [75].)

For the statement of the result (which will take place in the theorem below as well as in the following elaborations), we fix the realizations G(m, r, n) of the groups from family 2 as described in Section 11. Moreover we also fix the realization of the complex reflection groups G_{i} (4 ≤ i ≤ 37) to be the one described in [115]. Finally we let μ_{n} denote the group of n’th roots of unity.

Theorem 13.1 (Outer automorphisms of the exotic Z_{p}-reflection groups). Let (W, L) be an exotic Z_{p}-reflection group and let (W, V) be the associated complex reflection group. Then N_{GL(V)}(W) = ⟨W, C^{×}⟩ and hence N_{GL(L)}(W)/W = Z_{p}^{×}/Z(W) and N_{GL(L)}(W)/Z_{p}^{×}W = 1 except in the following cases:

(1) W = G(m, r, n) from family 2 with (m, r, n) ≠ (4, 2, 2), (3, 3, 3): N_{GL(V)}(W) = ⟨G(m, 1, n), C^{×}⟩ and N_{GL(L)}(W)/Z_{p}^{×}W = C_{gcd(r,n)}, cf. 13.4.

(2) W = G(4, 2, 2): N_{GL(V)}(W) = ⟨G_{8}, C^{×}⟩ and N_{GL(L)}(W)/Z_{p}^{×}W = Σ_{3}, cf. 13.5.

(3) W = G(3, 3, 3): N_{GL(V)}(W) = ⟨G_{26}, C^{×}⟩ and N_{GL(L)}(W)/Z_{p}^{×}W = A_{4}, cf. 13.6.

(4) W = G_{5}: N_{GL(V)}(W) = ⟨G_{14}, C^{×}⟩ and N_{GL(L)}(W)/Z_{p}^{×}W = C_{2}, cf. 13.7.

(5) W = G_{7}: N_{GL(V)}(W) = ⟨G_{10}, C^{×}⟩ and N_{GL(L)}(W)/Z_{p}^{×}W = C_{2}, cf. 13.8.

Lemma 13.2. Let K ⊆ K′ be fields of characteristic zero, and W ⊆ GL_{n}(K) an irreducible reflection group. Then N_{GL_{n}(K′)}(W) = ⟨N_{GL_{n}(K)}(W), K′^{×}⟩.

Proof. The inclusion “⊇” is clear, so suppose g ∈ N_{GL_{n}(K′)}(W). Consider the system of equations Xw = gwg^{−1}X, w ∈ W where X is an n × n-matrix. Over K′ this has the solution X = g. By [55, Lem. 2.10], the representation W → GL_{n}(K′) is irreducible, so the solution space is the 1-dimensional space spanned by g. Since the coefficients lie in K, the solution space over K is 1-dimensional as well, so we can write g = λg_{1} with λ ∈ K′ and g_{1} ∈ M_{n}(K). As g ≠ 0 we get λ ≠ 0 and g_{1} ∈ N_{GL_{n}(K)}(W). □

We can now start the proof of Theorem 13.1. By [21, Prop. 3.13] we easily get the results on N_{GL(V)}(W) claimed above (note that ⟨G(4, 1, 2), G_{6}, C^{×}⟩ = ⟨G_{8}, C^{×}⟩ and G(3, 1, 3) ⊆ G_{26}). Now assume that W does not belong to family 2 and W ≠ G_{5}, G_{7}. Let n denote the rank of W and K the field extension of Q generated by the entries of the matrices representing W. Our assumption ensures that N_{GL(V)}(W) = ⟨W, C^{×}⟩ and hence N_{GL_{n}(K)}(W) = ⟨W, K^{×}⟩. Lemma 13.2 now shows that N_{GL_{n}(Q_{p})}(W) = ⟨W, Q_{p}^{×}⟩. Hence we get N_{GL(L)}(W) = ⟨W, Z_{p}^{×}⟩ and since W is irreducible we have W ∩ Z_{p}^{×} = Z(W), cf. [55, Lem. 2.9].

This proves Theorem 13.1 in case W does not belong to family 2 and W ≠ G_{5}, G_{7}. In the remaining cases we have also proved the statements concerning N_{GL(V)}(W), and we thus only need to find the structure of N_{GL(L)}(W) in these cases. This is done in the Examples 13.4, 13.5, 13.6, 13.7 and 13.8 below.

To treat the dihedral group G(m, m, 2) from family 2 we need the following auxiliary result. Note also that the exotic groups from family 2a are also handled in [102, §6] (here the non-standard notation G(q, r; n) for G(q, q/r, n)).

Lemma 13.3. Assume that m ≥ 3 and p ≡ ±1 (mod m) so that ζ_{m} + ζ_{m}^{−1} ∈ Z_{p}. Then 2 + ζ_{m} + ζ_{m}^{−1} is a unit in Z_{p}.

Proof. It suffices to prove that the norm N_{Q(ζ_{m}+ζ_{m}^{−1})/Q}(2 + ζ_{m} + ζ_{m}^{−1}) is not divisible by p. Since its square equals the norm N_{Q(ζ_{m})/Q}(2 + ζ_{m} + ζ_{m}^{−1}) it is enough to see that this norm is not divisible by p. In Q(ζ_{m}) we have 2 + ζ_{m} + ζ_{m}^{−1} = (1 + ζ_{m})^{2}/ζ_{m} and since ζ_{m} is a unit it is enough to see that N_{Q(ζ_{m})/Q}(1 + ζ_{m}) is not divisible by p. By definition

N_{Q(ζ_{m})/Q}(1 + ζ_{m}) = ∏_{\begin{matrix}0≤k≤m \\ gcd(k,m)=1 \end{matrix}}(1 + ζ_{m}^{k}) = (−1)^{ϕ(m)} ∏_{\begin{matrix}0≤k≤m \\ gcd(k,m)=1 \end{matrix}}(−1 − ζ_{m}^{k}) = Φ_{m}(−1).

The first claim now follows from [127, Lem. 2.9]. □

Elaboration 13.4 (Family 2, generic case). Let W = G(m, r, n) from family 2 and let p be a prime number such that W is an exotic Z_{p}-reflection group. Thus if n ≥ 3 or n = 2 and r < m we have m ≥ 3 and p ≡ 1 (mod m), and for n = 2 and m = r we have m ≥ 5, m ≠ 6 and p ≡ ±1 (mod m). Assume moreover that (m, r, n) ≠ (4, 2, 2), (3, 3, 3) (these two cases will be dealt with below).

Assume first that p ≡ 1 (mod m). The realizations of the groups G(m, r, n) and G(m, 1, n) from above are both defined over the ring Z[ζ_{m}] which embeds in Z_{p}. Lemma 13.2 shows that N_{GL_{n}(Z_{p})}(W) = ⟨G(m, 1, n), Z_{p}^{×}⟩ whence the natural homomorphism (A(m, 1, n)/A(m, r, n))× Z_{p}^{×} ⟶ N_{GL_{n}(Z_{p})}(W)/W is surjective. The kernel is the cyclic group generated by the element ([ζ_{m}I_{n}], ζ_{m}^{−1}) (here [ζ_{m}I_{n}] ∈ A(m, 1, n)/A(m, r, n) denotes the coset of ζ_{m}I_{n}) and thus N_{GL_{n}(Z_{p})}(W)/W = (A(m, 1, n)/A(m, r, n)) ∘_{C_{m}} Z_{p}^{×}. Note that A(m, 1, n)/A(m, r, n) is cyclic of order r generated by the element x = [diag(1, …, 1, ζ_{m})] and that [ζ_{m}I_{n}] = x^{n}.

If the assumption p ≡ 1 (mod m) is not satisfied, then W = G(m, m, 2) is the dihedral group of order 2m with m ≥ 5, m ≠ 6 and p ≡ −1 (mod m). Conjugating the realization of G(m, m, 2) from above with the element

g = [ \begin{matrix}1 & −ζ_{m}^{−1} \\ 1 & −ζ_{m} \end{matrix} ]

gives a realization G(m, m, 2)^{g} defined over the character field Q(ζ_{m} + ζ_{m}^{−1}). Note that if m is odd, then N_{GL_{2}(C)}(G(m, m, 2) = ⟨G(m, 1, 2), C^{×}⟩ = ⟨G(m, m, 2), C^{×}⟩ and hence N_{GL_{2}(Z_{p})}(G(m, m, 2)^{g})/G(m, m, 2)^{g} = Z_{p}^{×}, so we may assume m to be even. Since G(m, 1, 2) is generated by G(m, m, 2) and diag(1, ζ_{m}) we find

N_{GL_{2}(Z_{p})}(G(m, m, 2)^{g}) = ⟨G(m, m, 2)^{g}, [ \begin{matrix}1 & 1 \\ −1 & 1 + ζ_{m} + ζ_{m}^{−1} \end{matrix} ] , Q_{p}^{×}⟩ ∩ GL_{2}(Z_{p})

using Lemma 13.2. From Lemma 13.3 we see that the above matrix is invertible over Z_{p} and hence

N_{GL_{2}(Z_{p})}(G(m, m, 2)^{g}) = ⟨G(m, m, 2)^{g}, [ \begin{matrix}1 & 1 \\ −1 & 1 + ζ_{m} + ζ_{m}^{−1} \end{matrix} ] , Z_{p}^{×}⟩

Thus the homomorphism Z × (Z_{p}^{×}/μ_{2}) ⟶ N_{GL_{2}(Z_{p})}(G(m, m, 2)^{g})/G(m, m, 2)^{g} which maps

(k, [λ]) to the coset of λ [ \begin{matrix}1 & 1 \\ −1 & 1 + ζ_{m} + ζ_{m}^{−1} \end{matrix} ]^{k} is surjective. The kernel is easily seen to

be the infinite cyclic group generated by the element (−2, [1 + ζ_{m} + ζ_{m}^{−1}]) and thus we get N_{GL_{2}(Z_{p})}(G(m, m, 2)^{g})/G(m, m, 2)^{g} ≅ Z∘_{Z}(Z_{p}^{×}/μ_{2}). It easily checked that [2+ζ_{m}+ζ_{m}^{−1}] has a square root in Z_{p}^{×}/μ_{2} if and only if either m ≡ 0 (mod 4) or m ≡ 2 (mod 4) and p ≡ −1 (mod 2m). In this case we have N_{GL_{2}(Z_{p})}(G(m, m, 2)^{g})/G(m, m, 2)^{g} ≅ C_{2} × (Z_{p}^{×}/μ_{2}).

Elaboration 13.5 (G(4, 2, 2)). The realization of the group G(4, 2, 2) from above and the realization of the group G_{8} from [115, p. 280–281] are both defined over their common character field Q(i). Thus the relevant primes p are the ones satisfying p ≡ 1 (mod 4). More precisely the representations are defined over Z[\frac{1}{2}, i] and as this ring embeds in Z_{p} for all p as above, we see that N_{GL_{2}(Z_{p})}(G(4, 2, 2)) = ⟨G_{8}, Z_{p}^{×}⟩. It is easily checked that G_{8} = ⟨G(4, 2, 2), H⟩, where H is the group of order 24 generated by the elements

[ \begin{matrix}0 & i \\ 1 & 0 \end{matrix} ] , \frac{1 + i}{2} [ \begin{matrix}1 & 1 \\ i & −i \end{matrix} ]

Since G(4, 2, 2) ∩ ⟨H, Z_{p}^{×}⟩ = Z(H) = μ_{4} we conclude that N_{GL_{2}(Z_{p})}(G(4, 2, 2))/G(4, 2, 2) ≅ (H/Z(H)) × (Z_{p}^{×}/μ_{4}) ≅ Σ_{3} × (Z_{p}^{×}/μ_{4}).

Elaboration 13.6 (G(3, 3, 3)). The realization of the group G(3, 3, 3) from above and the realization of the group G_{26} from [115, p. 296–297] are both defined over their common character field Q(ω) where ω = e^{2πi/3}. Thus the relevant primes p are the ones satisfying p ≡ 1 (mod 3). More precisely the representations are defined over Z[\frac{1}{3}, ω] and as this ring embeds in Z_{p} for all p as above, we see that N_{GL_{3}(Z_{p})}(G(3, 3, 3)) = ⟨G_{26}, Z_{p}^{×}⟩. It is easily checked that G_{26} is the semidirect product of G(3, 3, 3) with the group H ≅ SL_{2}(F_{3}) generated by the elements

R_{1} = [ \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & ω^{2} \end{matrix} ] , R_{2} = \frac{1}{\sqrt{−3}} [ \begin{matrix}ω & ω^{2} & ω^{2} \\ ω^{2} & ω & ω^{2} \\ ω^{2} & ω^{2} & ω \end{matrix} ]

The center of H is generated by the element

z = [ \begin{matrix}0 & −1 & 0 \\ −1 & 0 & 0 \\ 0 & 0 & −1 \end{matrix} ]

and G(3, 3, 3) ∩ ⟨H, Z_{p}^{×}⟩ = ⟨−z, μ_{3}⟩. Thus

N_{GL_{3}(Z_{p})}(G(3, 3, 3))/G(3, 3, 3) ≅ H ∘_{C_{2}} (Z_{p}^{×}/μ_{3}) ≅ SL_{2}(F_{3}) ∘_{C_{2}} (Z_{p}^{×}/μ_{3})

where the central product is given by identifying z ∈ H with the element in Z_{p}^{×}/μ_{3} represented by −1.

Elaboration 13.7 (G_{5}). The realization of the group G_{5} from [115, p. 280–281] is defined over the field Q(ζ_{12}), but the group has character field Q(ω) and thus the relevant primes p are the one satisfying p ≡ 1 (mod 3). Conjugation by the matrix

g = [ \begin{matrix}2 & \sqrt{3} − 1 \\ (\sqrt{3} − 1)(1 − i) & i − 1 \end{matrix} ]

gives a realization defined over Z[\frac{1}{3}, ω] which embeds in Z_{p} for all p as above. Its easily checked that G_{14} is generated by G_{5} and the reflection

S = \frac{1}{\sqrt{2}} [ \begin{matrix}−1 & i \\ −i & 1 \end{matrix} ]

From this we get

N_{GL_{2}(Z_{p})}(G_{5}^{g}) = ⟨G_{5}^{g}, [ \begin{matrix}0 & 1 \\ −2ω & 0 \end{matrix} ] , Z_{p}^{×}⟩

and thus the homomorphism Z × (Z_{p}^{×}/μ_{6}) ⟶ N_{GL_{2}(Z_{p})}(G_{5}^{g})/G_{5}^{g} which maps (k, [λ]) to the

coset of λ [ \begin{matrix}0 & 1 \\ −2ω & 0 \end{matrix} ]^{k} is surjective. The kernel is easily seen to be the infinite cyclic group

generated by the element (−2, [2]) and we get N_{GL_{2}(Z_{p})}(G_{5}^{g})/G_{5}^{g} ≅ Z ∘_{Z} (Z_{p}^{×}/μ_{6}). It is easy to check that [2] has a square root in Z_{p}^{×}/μ_{6} if and only if p ≡ 1, 7, 19 (mod 24) (that is unless p ≡ 13 (mod 24)). In this case we get the simpler description N_{GL_{2}(Z_{p})}(G_{5}^{g})/G_{5}^{g} ≅ C_{2} × (Z_{p}^{×}/μ_{6}).

Elaboration 13.8 (G_{7}). The realizations of the groups G_{7} and G_{10} given in [115, p. 280– 281] are both defined over their common character field Q(ζ_{12}). Thus the relevant primes p are the ones satisfying p ≡ 1 (mod 12). More precisely the representations are defined over Z[\frac{1}{2}, ζ_{12}] and as this ring embeds in Z_{p} for all p as above, we see that N_{GL_{2}(Z_{p})}(G_{7}) = ⟨G_{10}, Z_{p}^{×}⟩. It is easily checked that G_{10} = ⟨G_{7}, C_{4}⟩, where C_{4} is the cyclic group generated

by [ \begin{matrix}1 & 0 \\ 0 & i \end{matrix} ]. Since G_{7} ∩ (C_{4} × Z_{p}^{×}) = C_{2} × μ_{12} we conclude that N_{GL_{2}(Z_{p})}(G_{7})/G_{7} ≅

C_{2} × (Z_{p}^{×}/μ_{12}).

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Department of Mathematics, University of Copenhagen, 2100 Copenhagen, Denmark

   E-mail address: kksa@math.ku.dk

Department of Mathematics, University of Chicago, Chicago, IL 60637, USA

   E-mail address: jg@math.uchicago.edu

Department of Mathematics, University of Copenhagen, 2100 Copenhagen, Denmark

   E-mail address: moller@math.ku.dk

Department of Mathematics, University of Malaga, 29080 Malaga, Spain

   E-mail address: viruel@agt.cie.uma.es