FOR WHICH PSEUDO REFLECTION GROUPS

ARE THE p–ADIC POLYNOMIAL INVARIANTS

AGAIN A POLYNOMIAL ALGEBRA?

D. Notbohm

Abstract. Let W be a finite group acting on lattice L over the p–adic integers ℤ_{p}^{∧}. The analysis of the ring of invariants of the associated W –action on the algebra ℤ_{p}^{∧}[L] of polynomial functions on L is a classical question of invariant theory. If p is coprime to the order of W, classical results show that W is a pseudo reflection group, if and only if the ring of invariants is again polynomial. We analysis the situation for those odd primes dividing the order of W and, in particular, determine those pseudo reflection groups for which the ring of invariants ℤ_{p}^{∧}[L]^{W} is a polynomial algebra.

1. Introduction.

Let L be a p–adic lattice, i.e. a torsion free finitely generated ℤ_{p}^{∧}–module, and let ℤ_{p}^{∧}[L] be the graded polynomial algebra of polynomial functions on L. Following the conventions of topology we give the elements of the dual L^{♯} := Hom_{ℤ_{p}^{∧}}(L, ℤ_{p}^{∧}) ⊂ ℤ_{p}^{∧}[L] the degree 2. Every (faithful) representation W ⟶ Gl(L) of a finite group W establishes an W –action on ℤ_{p}^{∧}[L]. The analysis of the ring of invariants is a classical question of invariant theory. In particular, one might ask, whether ℤ_{p}^{∧}[L]^{W} is again a polynomial algebra. If this is the case, we call the lattice L polynomial (with respect to the W –action).

If we work over a field of characteristic coprime to the order |W | of the group W, the Shepard–Todd–Chevalley–Theorem [11] [4] says that, if W is a pseudo reflection group, then the invariants are again a polynomial algebra. And the converse is also true (e.g. see [12; Theorem 7.4.1].

For a vector space V, a representation W ⟶ Gl(V) of a finite group is called a pseudo reflection group if it is faithful and if the image is generated by pseudo reflection; i.e. by elements of finite order which fix a hyperlane of V of codimension 1. For actions on p–adic lattices the same definition works. If we say that W is a pseudo reflection group, then we always have a particular representation in mind. Examples of p–adic pseudo reflection groups are given by the action of the Weyl group W_{G} of a connected compact Lie group G on a maximal torus T_{G} of G, actually on the p–adic lattice L_{G} := H_{2}(BT_{G}; ℤ_{p}^{∧}).

If p does not divide |W |, then, as an easy consequence, the Shepard–Todd– Chevalley Theorem as well as it’s converse also holds for p–adic integral representations; i.e. a representation W ⟶ Gl(L) is a pseudo reflection group if and only if the lattice L is polynomial (see Corollary 3.6 and Proposition 3.7).

Typeset by AMS-TEX

In the modular case, that is p divides |W |, only the converse is true: if ℤ_{p}^{∧}[L]^{W} is a polynomial algebra, then W ⟶ Gl(L) is a pseudo reflection group(see Proposition 3.7).

In this work, we are mainly interested in the question, for which p–adic pseudo reflection groups the Shepard-Todd–Chevalley–Theorem holds; i.e. for which pseudo reflection groups W ⟶ Gl(L) the W –lattice L is polynomial. For odd primes, we will give a complete answer to this question, in particular in the modular case.

Before we can state our results explicitely we first have to make some definitions and to fix notation.

1.1 Definitions and remarks. 1.1.1 Let L be a torsionfree ℤ_{p}^{∧}–module. Every p–adic representation W ⟶ Gl(L) gives rise to an representation W ⟶ Gl(L_{ℚ}) over ℚ_{p}^{∧}, where L_{ℚ} := L ⊗_{ℤ_{p}^{∧}} ℚ. This p-adic rational representation might contain several W –lattices. From this point of view that are lattices of L_{ℚ} which are stable under the action of W.

Several notions of L_{ℚ} are inherited to L; e.g. the represetation W ⟶ Gl(L) is called irreducible if W ⟶ Gl(L_{ℚ}) is irreducible.

Two W –lattices L_{1} and L_{2} are called isomorphic if L_{1} and L_{2} are isomorphic as modules over the group ring ℤ_{p}^{∧}[W]. They are called weakly isomorphic if there exists an automorphims α : W ⟶ W such that L_{1} and L_{2}^{α} are isomorphic. Here, the action of W on L_{2}^{α} is given by the composition of ρ_{2} and α. 1.1.2 Let L be p–adic lattice and let W ⟶ Gl(L) be a pseudo reflection group. The covariants L_{W} := L/SL are defined as the quotient of L by SL, where SL ⊂ L is sublattice generated by all elements of the form l − w(l) with l ∈ L and w ∈ W. The lattice L is called simply connected if L_{W} = 0. If p is odd, the lattice SL is always simply connected. [8; 3.2]. 1.1.3 For every W –lattice L, there exists a short exact sequence

0 ⟶ L ⟶ L_{ℚ} ⟶ L_{∞} := L_{ℚ}/L ⟶ 0

of W –modules. The quotient L_{∞} ≅ (ℤ/p^{∞})^{n} ⊂ (S^{1})^{n} is called a p–discrete torus and can be considered as a subgroup of a torus whose dimension equals the rank of L.

For a p–discrete torus L_{∞} with an action of W, we get an W –lattice by setting L := Hom(ℤ/p^{∞}, L_{∞}). Infact, L is a lattice because ℤ_{p}^{∧} ≅ Hom(ℤ/p^{∞}, ℤ/p^{∞}). For every W –lattice L, we always have L ≅ Hom(ℤ/p^{∞}, L_{∞}). 1.1.4 Given a short exact sequence L ⟶ M ⟶ K of W –modules, such that L and M are lattices and such that K is finite, the two reperesentation L_{ℚ} and M_{ℚ} of W are isomorphic. Thus, the above short exact sequence of 1.1.2 and the serpent lemma establish a short exact sequence K ⟶ L_{∞} ⟶ M_{∞}.

In [8] the following structure theorem is proved:

1.2 Theorem. Let p be an odd prime. Let L be a p–adic lattice and let W ⟶ Gl(L) be a pseudo reflection group.

i) The W –lattice L fits into two short exact sequences of W –modules, namely

SL ⟶ L ⟶ L_{W}

and

SL ⊕ L^{W} ⟶ L ⟶ K ,

where SL is simply connected and where W acts trivially on on the finite W –module K ≅ (L/L^{W})_{W} ≅ L_{W} /L^{W}. ii) Let S be a simply connected W –lattice. Then, there exist splittings W = ∏_{i} W_{i} and S ≅ ⨁_{i} S_{i} in such a way that W_{j} acts trivially on S_{i} if i ≠ j and S_{i} is a simply connected irreducible W_{i} lattice. iii) If, in addition, L is an irreducible W –lattice, then, up to weak isomorphism, there exists a unique simply connected W –lattice S ⊂ L_{ℚ}, given by the SL.

Proof. This follows from [8; theorems 1.2, 1.3, 1.4]. The small differences in the statements come from the slightly different definition of W –lattices. □

We can now state our main results.

1.3 Theorem. Let p be an odd prime. A W –lattice L is polynomial if and only if SL is polynomial and L_{W} is torsion free if and only if SL is polynomial and the composition K ⟶ SL_{∞} ⊕ (L^{W})_{∞} ⟶ (L^{W})_{∞} is a monomorphism.

This result and the second part of Proposition 1.2 reduces the question of polynomial W –lattices to one about simply connected irreducible W –lattices.

1.4 Theorem. Let p be an odd prime. Let W ⟶ Gl(U) be an irreducible pseudo reflection group over ℚ_{p}^{∧}. Let S ⊂ U be a simply connected W –lattice of U. Then, S is polynomial if and only if the pair (W, p) does not belong to one of the pairs (W_{F_{4}}, 3), W_{E_{6}}, 3), W_{E_{7}}, 3), (W_{E_{8}}, 3) and (W_{E_{8}}, 5).

The pairs excluded by the theorem are all given by the Weyl group action of exceptional connected compact Lie groups G on L_{G} at those odd primes which appear as torsion primes in the cohomology of G.

In [5] is given a complete list of all irreducible p–adic rational pseudo reflection groups. In particular, Theorem 1.4 says that for all other cases the simply connected W –lattices are polynomial.

Starting from a simply connected polynomial W –lattice S, we can construct a polynomial lattice in the following way: We choose a lattice Z with trivial W – action, a finite subgroup K ⊂ (S_{∞})^{W} and a monomorphism K ⟶ Z_{∞}. Then we define L_{∞} := (S_{∞} ⊕ Z_{∞})/K and L := Hom(ℤ/p^{∞}, L_{∞}). By construction and because Ext(ℤ/p^{∞}, K) ≅ K, we get a short exact sequence S ⊕ Z ⟶ L ⟶ K of W –modules. Moreover, one can show that Z ≅ L^{W} (see Lemma 2.2). By Theorem 1.3, the lattice L is polynomial and every polynomial lattice can be constructed this way.

The proof of Theorem 1.4 is very much based on the classifictaion of W –lattices which is only known for odd primes [8]. It also uses the classification of irreducible p-adic rational pseudo reflection groups [5] and is done by ckecking case by case. For the proof of Theorem 1.3 we use a functorial classifying space construction to produce a fibration B^{2}SL ⟶ BL ⟶ B^{2}(L_{W}) which W acts on. The associated Serre spectral sequence carries an W –action and turns out to be compatible with taking fixed–points. This is the key of the proof. At one place we also need that, for odd primes, the kernel of Gl(n; ℤ_{p}^{∧}) ⟶ Gl(n; 𝔽_{p}) is torsionfree. That is, where the restriction to odd primes comes from.

The paper is organized as follows. In Section 2 we recall some material about W –lattices from [nolattice] and in Section 3 we discuss sufficient conditions for a W –lattice being polynomial. The last two section are devoted to the proof of Theorem 1.3 and Theorem 1.4.

Beside the algebraic motivation there is one reason why algebraic topologists might be interested in questions of this type. A classical question of Steenrod [13] asks for polynomial algebras over the field 𝔽_{p} which can be realized as the mod–p cohomology of a space. Work of Adams and Wilkerson [1] and Dwyer, Miller and Wilkerson [6] show that, at least for odd primes, all those algebras are isomorphic to ℤ_{p}^{∧}[L]^{W} ⊗ 𝔽_{p} for a polynomial lattice L of a pseudo reflection group W ⟶ Gl(L_{ℚ}). Based on these results, for odd primes, we will give a complete answer to Steenrod’s question in a further paper [9].

2. Lattices of pseudo reflection groups.

In this section we recall some further material from [8] about p–adic representations of pseudo reflection groups. We always assume that p is an odd prime.

Let U be a finite dimensional vector space over ℚ_{p}^{∧}. Let W ⟶ Gl(U) be pseudo reflection group.

2.1 Definitions and remarks.

2.1.1 In 1.1.3 we desribed an algebraic way of constructing a lattice from a p–discrete torus. There is another, a topological way to do this. Let L_{∞} be a p–discrete torus with an action of a finite group W. Then, using a functorial classifying space construction, we can construct a W –lattice by L := H_{2}(BL_{∞}; ℤ_{p}^{∧}). If we start with an W –lattice L, then we have equivalences L ≅ H_{2}(BL_{∞}; ℤ_{p}^{∧}), L ≅ H_{2}(B^{2}L; ℤ_{p}^{∧}) and ℤ_{p}^{∧}[L] ≅ H^{∗}(B^{2}L; ℤ_{p}^{∧}) (see [8; 1.1]).

Moreover, every short exact sequenz L ⟶ M ⟶ N of W –lattices establishes an fibration B^{2}L ⟶ B^{2}M ⟶ B^{2}N of W –equivariant spaces.

2.1.2 For abbreviation we set L/p := L ⊗ 𝔽_{p}. Analogously as above we have 𝔽_{p}[L/p] ≅ H^{∗}(B^{2}L; 𝔽_{p}) ≅ ℤ_{p}^{∧}[L] ⊗ 𝔽_{p}. The last isomorphism is obvious.

2.1.3 A monomorphism L ⟶ M of W –lattices is called a W –trivial restriction or W –trivial extension if W acts trivially on the quotient M/L and if M/L is finite.

2.2 Lemma. Let S ⊕ Z ⟶ L ⟶ K be a W –tivial restriction of a pseudo reflection group W ⟶ Gl(L). If S is simply connected and W acts trivial on Z, then Z ≅ L^{W}.

Proof. The quotient L/Z is rationally isomorphic to S and therefore fixed–point free. Taking fixed points gives an exact sequence 0 ⟶ Z ⟶ L^{W} ⟶ (L/Z)^{W} = 0. □

3. Invariant theory for p–adic pseudo reflection groups.

In this section we discuss sufficent conditions, that, for a given pseudo reflection group W ⟶ Gl(L), the W –lattice L or the vector space L/p are polynomial.

The following well known result might be found in [12; 5.5.5].

3.1 Proposition. Let V be an n–dimensional vector space over a field 𝔽. Let W ⟶ Gl(V) be a faithful representation of a finite group. For i = 1, ..., n, let g_{i} ∈ 𝔽[V]^{W}. Then the quotient 𝔽[V]/(g_{1}, ...g_{n}) is a finite dimensional vector space over 𝔽 and ∏_{i} deg(g_{i})/2 = |W | if and only if 𝔽[V]^{W} is a polynomial algebra generated by g_{1}, ...., g_{n}.

We also need a p–adic version of this statement for lattices.

3.2 Corollary. Let p be an odd prime. Let L ≅ ℤ_{p}^{∧}^{n} be a p–adic lattice carrying a faithful action of a finite group W. For i = 1, ..., n, let g_{i} ∈ ℤ_{p}^{∧}[L]^{W}. Then ∏_{i} deg(g_{i})/2 = |W | and the quotient ℤ_{p}^{∧}[L]/(g_{1}, ...g_{n}) is a finitely generated module over ℤ_{p}^{∧} if and only if ℤ_{p}^{∧}[L]^{W} is a polynomial algebra generated by g_{1}, ...., g_{n}.

Proof. Let A := ℤ_{p}^{∧}[L]. Let : A ⟶ A/p ≅ 𝔽_{p}[L/p] denote the reduction mod p. Since p is odd and since W is a finite group the representation W ⟶ Gl(L/p) is faithful and we can apply Proposition 3.1 for the W –action on L/p.

If A/(g_{1}, ..., g_{n}) is a finitely generated ℤ_{p}^{∧}–module, the quotient (A/p)/(\overline{g}_{1}, ..., \overline{g}_{n}) is finite dimesional 𝔽_{p}–vector space. Since the degree equation is independent of the basic ring, this shows that A/p^{W} ≅ 𝔽_{p}[\overline{g}_{1}, ..., \overline{g}_{n}] ≅ A^{W} /p is a polynomial algebra generated by the elements \overline{g}_{i} (Proposition 3.1), and, by the Nakayama lemma, that A^{W} is also a polynomial algebra generated by the elements g_{i} for i = 1, ...n.

If A^{W} ≅ ℤ_{p}^{∧}[g_{1}, ..., g_{n}] then A/(g_{1}, ...g_{n}) is a lattice in the finite dimensional ℚ_{p}^{∧}–vector space A ⊗ ℚ/(g_{1}, ..., g_{n}) and therefore a finitely generated ℤ_{p}^{∧}–module. The degree equation also follows from Proposition 3.1 for 𝔽 = ℚ_{p}^{∧}. □

3.3 Lemma. Let p be an odd prime. Let L be a W –lattice. Then the following holds:

i) If L is polynomial, the map ℤ_{p}^{∧}[L]^{W} ⟶ 𝔽_{p}[L/p]^{W} is an epimorphism and 𝔽_{p}[L/p]^{W} is a polynomial algebra. ii) If 𝔽_{p}[L/p]^{W} is polynomial and ℤ_{p}^{∧}[L]^{W} ⟶ 𝔽_{p}[L/p]^{W} is an epimorphism, then L is polynomial.

Proof. Let A := ℤ_{p}^{∧}[L]. Let g_{1}, ..., g_{n} be polynomial generators of A^{W}. Then we have ∏_{i} deg(g_{i})/2 = |W | (Corollary 3.2).

The sequence A^{W} ⟶ A ⟶ A/A^{W} is short exact and splits, since A/A^{W} is torsionfree, and A/A^{W} is a finitely generaterd ℤ_{p}^{∧}–module (Corollary 3.2). Reducing mod p establishes a short exact sequence A^{W} /p ⟶ A/p ≅ 𝔽_{p}[L/p] ⟶ (A/A^{W})/p ≅ (A/p)/(A^{W} /p). The third term is a finite dimensional 𝔽_{p}–vector space. Let \overline{g}_{i} be the image of g_{i} under the reduction. Then the elements \overline{g}_{i} ∈ A/p are invariants and generate A^{W} /p. Therefore, (A/p)/(\overline{g}_{1}, ..., \overline{g}_{n}) ≅ (A/p)/(A^{W} /p) is finite dimensional. As in the proof of Corollary 3.2 we can apply Proposition 3.1. Hence, because ∏_{i} deg(\overline{g}_{i})/2 = ∏_{i} deg(g_{i})/2 = |W |, the ring (A/p)^{W} of invariants is a polynomial algebra generated by \overline{g}_{1}, ..., \overline{g}_{n} (Proposition 3.1) and the map ℤ_{p}^{∧}[L]^{W} ⟶ 𝔽_{p}[L/p]^{W} is an epimorphism. This proves the first part.

Let 𝔽_{p}[L/p]^{W} ≅ 𝔽_{p}[\overline{g}_{1}, ..., \overline{g}_{n}] be a polynomial algebra. Let g_{i} ∈ ℤ_{p}^{∧}[l]^{W} be a lift of \overline{g}_{i}. Then, by the Nakayama lemma, the elements g_{i} generate A^{W} as a polynomial algebra. □

3.4 Remark. If, for a W –lattice L, the reduction L/p is polynomial, it does not follow that L itself is polynomial. The following example, demonstrating this fact, was pointed to us by A. Viruel.

Let p = 3 and G = SU(3)/ℤ/3. Then, the action of W_{G} on L_{G} := H_{2}(BT_{G}; ℤ_{p}^{∧}) makes L_{G}/p polynomial. The ring of invariants 𝔽_{p}[L_{G}/p]^{W_{G}} is generated by two elements of degree 2 and 12 [3]. Since rationally there is no differenz between G = SU(3)/ℤ/3 and SU(3), the polynomial algebra (ℤ_{p}^{∧}[L_{G}] ⊗ ℚ)^{W_{G}} is generated by two elements of degree 4 and 6. Thus L_{G} can’t be polynomial, since otherwise reduction mod p would establish an epimorphisms between the invariants (Lemma 3.3).

The following statement is a relative version of the theorem of Shepard and Todd [11] respectively of Chevalley [4].

3.5 Proposition. Let V be vector space over a field 𝔽. Let W ⟶ Gl(V) be a pseudo reflection group. Let W_{1} ⊂ W be a subgroup such the index [W : W_{1}] is coprime to the characteristic char(𝔽) of 𝔽. If 𝔽[V]^{W_{1}} is a polynomial algebra, then so is 𝔽[V]^{W}.

Proof. By [12; 6.4.4] it is sufficient to show that 𝔽[V]^{W_{1}} is a free 𝔽[V]^{W} –module, and by [12; 6.1.1] this follows if T or_{1}^{𝔽[V]^{W}} (𝔽, 𝔽[V]^{W_{1}}) vanishes. Since ([W : W_{1}], char(𝔽)) = 1, there exists an averaging map

a : 𝔽[V]^{W_{1}} ⟶ 𝔽[V]^{W} : f ↦ (1/[W : W_{1}]) ∑_{w∈W/W_{1}} wf .

The vanishing of the T or_{1}–term now follows analogously as in the proof of the Shepard–Todd theorem as given in [12; 7.4.1]. □

Actually, for our purpose, we need a p–adic integral version of Proposition 3.5.

3.6 Corollary. Let L be W –lattice. Let W ⟶ Gl(L) be a pseudo reflection group. Let W_{1} ⟶ W be a subgroup of index coprime to p. If L is polynomial as W_{1}–lattice, then so is L as W –lattice.

Proof. Since the index [W : W_{1}] is a unit in ℤ_{p}^{∧}, there exists again the averaging map a : ℤ_{p}^{∧}[L]^{W_{1}} ⟶ ℤ_{p}^{∧}[L]^{W} which, in particular, is an epimorphism. The same holds after reducing everything mod p. Therefore, in the diagram

ℤ_{p}^{∧}[L]^{W_{1}} ⟶ ℤ_{p}^{∧}[L]^{W}
↓  ↓
𝔽_{p}[L/p]^{W_{1}} ⟶ 𝔽_{p}[L/p]^{W}

the bottom and the left vertical arrow are epimorphisms (Lemma 3.3) as well as the right vertical map. By Lemma 3.3 follows that L is a polynomial W –lattice. □

If we choose the trivial group for W_{1} in the the last two statements, we get the Shepard–Todd–Chevalley Theorem. We end this section with an p–adic integral version of the converse.

3.7 Proposition. Let L be a finitely generated lattice and W ⟶ Gl(L) be a faithful representation of a finite group of a finite group W. If ℤ_{p}^{∧}[L]^{W} is a polynomial algebra, then W ⟶ Gl(L) is a pseudo reflection group.

Proof. Because ℚ_{p}^{∧}[L_{ℚ}]^{W} ≅ ℤ_{p}^{∧}[L]^{W} ⊗ ℚ, both are polynomial algebras. Now, applying the Shepard–Todd–Chevalley Theorem in characteristic 0 shows that W ⟶ Gl(L_{ℚ}) as well as W ⟶ Gl(L) are pseudo reflection groups.

4. Proof of Theorem 1.3.

For the proof of Theorem 1.3 we need the following results.

4.1 Lemma. Let G be a finite group and let M be a W –lattice. Then, M^{G} ⟶ (M/p)^{G} is an epimorphism if and only if H^{1}(G; M) = 0.

Proof. Multiplication by p gives rise to a short exact sequence M ⟶ M ⟶ M/p. Taking fixed–points establishes an exact sequence M^{G} ⟶ M^{G} ⟶ (M/p)^{G} ⟶ H^{1}(G; M) ⟶ H^{1}(G; M). The first and last map are again given by multiplication by p. Since M is a ℤ_{p}^{∧}–module and since G is finite, the group H^{1}(G; M) is a finite abelian p group. Therefore, the second map in the above sequence is an epimorphism iff H^{1}(G; M) vanishes. □

4.2 Corollary. Let L be polynomial W –lattice. Then, H^{1}(W; ℤ_{p}^{∧}[L]) = 0.

Proof. Since L is polynomial, the map ℤ_{p}^{∧}[L]^{W} ⟶ 𝔽_{p}[L/p]^{W} is an epimorphism (Lemma 3.3) □

Now we can start with the proof of Theorem1.3.

Proof of Theorem 1.3. Let us first assume that S := SL is polynomial and that Z := L_{W} is torsionfree. The short exact sequence S ⟶ L ⟶ Z establishes a fibration B^{2}S ⟶ B^{2}L ⟶ B^{2}Z (see 2.1.1) of W –equivariant spaces, where W acts trivially on the last term. All spaces are classifying spaces of p–adic tori and H^{∗}(B^{2}S; ℤ_{p}^{∧}) ≅ ℤ_{p}^{∧}[S] (analogously for the two other lattices). Now we look at the associated W –equivariant Serre spectral sequence whose E_{2}–term is given by E_{2}^{∗,∗} ≅ H^{∗}(B^{2}Z; ℤ_{p}^{∧}) ⊗ H^{∗}(B^{2}S; ℤ_{p}^{∧}). By degree reasons all differentials vanish. The extension problems are given by short exact sequences of the form F^{∗,∗} ⟶ F^{∗+1,∗−1} ⟶ H^{∗}(B^{2}Z; ℤ_{p}^{∧}) ⊗ H^{∗}(B^{2}S; ℤ_{p}^{∧}). We claim that, taking fixed–points, gives again a short exact sequence. That is we have to show that H^{1}(W; F^{∗,∗}) vanishes for all (∗, ∗). But this follows via an induction based on the above exact sequence from Corollary 4.2.

In particular, this argument shows that H^{∗}(B^{2}L; ℤ_{p}^{∧})^{W} ⟶ H^{∗}(B^{2}S; ℤ_{p}^{∧})^{W} is an epimorphism and that H^{∗}(B^{2}Z; ℤ_{p}^{∧}) ⟶ H^{∗}(B^{2}L; ℤ_{p}^{∧})^{W} is a monomorphism. Let g_{1}, ..., g_{n} be polynomial generators of H^{∗}(B^{2}S; ℤ_{p}^{∧})^{W}. Choosing lifts of this classes in H^{∗}(B^{2}L; ℤ_{p}^{∧})^{W} establishes an isomorphism H^{∗}(B^{2}Z; ℤ_{p}^{∧}) ⊗ H^{∗}(B^{2}S; ℤ_{p}^{∧})^{W} ≅ H^{∗}(B^{2}L; ℤ_{p}^{∧})^{W}, which shows that L is polynomial.

Now let us assume that L is polynomial. We first want to show that L_{W} is torsionfree. The lattice L fits into an W –trivial extension S ⊕ Z ⟶ L ⟶ K with Z ≅ L^{W} (Thereom 1.2). We dualize this sequence, i.e. we apply the functor Hom( , ℤ_{p}^{∧}). Let L^{♯} denote the dual. We get a short exact sequence

L^{♯} ⟶ S^{♯} ⊕ Z^{♯} ⟶ Ext(K, ℤ_{p}^{∧}) ≅ K .

Taking fixed–points yields the exact sequence given by the top row of the commutative diagram

0 ⟶ L^{♯}^{W} ⟶ S^{♯}^{W} ⊕ Z^{♯} ≅ Z^{♯} ⟶ K ⟶ H^{1}(W; L^{♯})
↓  ↓  ∥
0 ⟶ L^{♯} ⟶  S^{♯} ⊕ Z^{♯}  ⟶ K ⟶  0
↓  ↓  ↓
0 ⟶ L^{♯}/L^{♯}^{W} ⟶^{≅}  S^{♯}  ⟶ 0  .

The isomorphism in the middle of the top row follows since S as well as S^{♯} are fixed–point free. AAll rows and columns are exact.

Having in mind that, for a lattice M, we have (M^{♯})^{♯} ≅ M and dualizing the first column establishes the short exact sequence

S ⟶ L ⟶ ((L^{♯})^{W})^{♯} ≅ L_{W} .

Therefore, the module L_{W} is torsionfree.

Now we want to show that S is polynomial. Let A_{L} := ℤ_{p}^{∧}[L] and define A_{S} analogously. By assumption we have A_{L}^{W} ≅ ℤ_{p}^{∧}[f_{1}, ...f_{r}, g_{1}, ...g_{s}] for suitable elements f_{i} and g_{j}, where deg(f_{i}) = 2 and deg(g_{j}) > 2. The number of generators are determined by the dimension of the lattices. That is that s + r = dim(L) . Since S^{♯} is fixed–point free, the elements f_{i} are images of elements of Z^{♯} and r ≤ dimZ. Since Z^{♯} ⊂ L^{♯}^{W} we have r ≥ dim(Z) and therefore r = dim(Z) and s = dim(S). Now let g′_{j} ∈ A_{S}^{W} be the images of g_{j} ∈ A_{L}^{W}. Then ∏_{j} deg(g′_{j})/2 = ∏_{j} deg(g_{j})/2 ∏_{i} deg(f_{i})/2 = |W | (Corollary 3.2). Moreover, the map A_{L}/(f_{i}, g_{j}) ⟶ A_{S}/(g′_{j}) is an epimorphism and therefore, the target is also finitely generated. Applying Corollary 3.2 again yields A_{S}^{W} ≅ ℤ_{p}^{∧}[g_{j}]. In particular, the lattice S is polynomial. This finishes the proof of the first equivalence of statements. The second follows from the lemma below. □

4.3 Lemma. Let S ⊕ Z ⟶ L ⟶ K be a W –trivial extension, such that W acts trivially on Z and such that S is simply connected. Then L_{W} is torsionfree if and only if the composition K ⟶ S_{∞} ⊕ Z_{∞} ⟶ Z_{∞} is injective.

Proof. Since S is simply connected, passing to covariants establishes a short exact sequence Z ⟶ L_{W} ⟶ K. If L_{W} is torsionfree, this is a W –trivial extension and establishes the short exact sequence K ⟶ Z_{∞} ⟶ (L_{W})_{∞}. In particular, K ⟶ Z_{∞} is a monomorphism.

If K ⟶ Z_{∞} is a monomorphism the quotient Z′_{∞} := Z_{∞}/K defines a lattice Z′ := Hom(ℤ/p^{∞}, Z′_{∞}) with trivial W –action and fits into the diagram

S_{∞}  S_{∞}
↓  ↓
K ⟶ S_{∞} ⊕ Z_{∞} ⟶ L_{∞}
∥  ↓  ↓
K ⟶ Z_{∞} ⟶ Z′_{∞} .

Passing to W –lattices, i.e. applying the functor Hom(ℤ/p^{∞}, ) shows that L_{W} ≅ Z′ is torsionfree. □

5. Proof of Theorem 1.4.

proof of Theorem 1.4. Let W ⟶ Gl(n; ℚ_{p}^{∧}) be an irreducible pseudo reflection group over the p–adic rationals. There exists a complete classification of these objects (see [5]). Let S ⊂ ℚ_{p}^{∧}^{n} be the simply connected W lattice unique up to isomorphism (see Theorem 1.2).

For the pairs (W, p) listed in the statement, the prime p is always a torsion prime of the integral cohomology of the connected compact Lie group G. Thus, the simply connected W_{G}–lattices H_{2}(BT_{G}; ℤ_{p}^{∧}) are not polynomial. This proves one half of the statement.

We now show that, for all other pairs (W, p), the associated simply connected lattice S is polynomial. If (p, |W |) = 1, we can apply Corollary 3.6 to see that S is polynomial. Checking the complete list of irreducible pseudo reflection groups over ℚ_{p}^{∧} of Clark and Ewing [5], there are only a few cases left. For all these cases, there exists a connected compact Lie group G such that W_{G} ⊂ W ⟶ Gl(S) is a W_{G} lattice isomorphic to L_{G} := H_{2}(BT_{G}; ℤ_{p}^{∧}) and such that L_{G} is polynomial. Here, T_{G} ⊂ G denotes a maximal torus of G. The particular information about the Lie groups, the subgroups and the primes are given in the following table. The numbering refers to the Clark–Ewing list [5].

No  G  W_{1}  W  prime
1  SU(n)  Σ_{n}  Σ_{n}  p ≤ n
2a  U(n)  Σ_{n}  W ⊂ ℤ/r ≀ Σ_{n}  r|(p − 1), p ≤ n
2b  SU(3)  Σ_{3}  D_{12}, D_{6}  p = 3
12  SU(3)  Σ_{3}  Gl(2; 𝔽_{3})  p = 3
29  SU(5)  Σ_{5}  W  p = 5
31  SU(5)  Σ_{5}  W  p = 5
34  SU(7)  Σ_{7}  W  p = 7

For all other cases, the pair (W, p) belongs to the excluded list or (p, |W |) = 1. The information given in this table is obvious for No. 1 and may be found in [10] for No. 2a and in [2] for all other cases. For all cases besides No. 2a, S is already simply connected as W_{1}–lattice and therefore simply connected as W –lattice. For No. 2a, we have (S^{♯}/p)^{W} = 0 [10; Proposition 1.4]. Hence, by [8; Lemma 2.2 and Proposition 5.1], S is simply connected as W –lattice.

Now we can apply Corollary 3.6, which shows that S is a polynomial W –lattice. This finishes the proof of Theorem 1.4. □

From the mentioned references [10] and [2], for the lattices of the above table, you can get a little extra piece of information about the subgroup W_{1} ⊂ W. Actually, if Δ ≅ ℤ/p ⊂ L/p describes the diagonal of L/p, then W_{1} ≅ Iso_{W} (Δ) is given as the isotropy subgroup of Δ.

For (p, |W |) = 1, we define Δ := L/p. Then, W_{1} := Iso_{W} (Δ) is the trivial group (p is odd). Taking the trivial connected compact Lie group as G we get the same picture as desribed in the above table. This shows that the following proposition is true at least for simply conneceted irreducible W –lattices.

5.1 Proposition. Let p be an odd prime. Let L be a polynomial W –lattices. Then, there exists a compact connected Lie group G and a subspace Δ ⊂ L/p such that the following holds:

i) W_{G} ⊂ W ⟶ Gl(L) is isomorphic to the W_{G}–lattice L_{G}. ii) L is a polynomial W_{G}–lattice and ℤ_{p}^{∧}[L]^{W_{G}} ≅ H^{∗}(BG; ℤ_{p}^{∧}). iii) W_{G} = Iso_{W} (Δ) and the index [W : W_{G}] is coprime to p.

Proof. The statement is true for simply connected W –lattices, since these split into a product of irreducible simply connected lattices.

Every polynomial lattice L fits into an exact sequence SL × Z ⟶ L ⟶ K, where W acts trivially on Z and K, such that SL is simply connected and polynomial and such that the associated monomorphism K ⟶ Z_{∞} is a monomorphism (Theorem 1.3). Now, we choose Δ ⊂ SL/p and a connected compact Lie group G which satisfy the statement for SL. Since the rpresentation W_{G} ⟶ Gl(L_{G}) has an integral lift, the inclusion SL_{∞} ⊂ T_{G} is W –equivariant, and K ⊂ T_{G}^{W_{G}}. By the lemma below, the index of the center Z(G) ⊂ T_{G}^{W_{G}} is a power of 2. Hence, K ⊂ T_{G} ⊂ G is a central subgroup.

We also can realize Z as the p–adic lattice of an integral torus T with trivial W_{G}–action. Let H := (G × T)/K. Then, W_{H} ≅ W_{G} ⊂ W has index copriome to p and, by construction, the composition W_{H} ⊂ W ⟶ Gl(L) is isomorphic to the W_{H}–lattice L_{H}. Moreover, Δ ⊂ L/p and W_{H} ≅ Iso_{W} (Δ). This shows that the statement is true for general polynomial W –lattices. □

The following lemma might be well known, but we couldn’t find a reference for it stating the result in terms of compact Lie groups.

3.2 Lemma. For any connected compact Lie group G, the index [T_{G}^{W_{G}} : Z(G)] is a power of 2.

Proof. We only have to show that, for odd primes every p–toral subgroup P ⊂ T_{G}^{W_{G}} is already contained in Z(G). Let C ⊂ G be the centralizer of P. Then, C is of maximal rank, the weyl groups W_{C} = W_{G} are identical and π_{0}(C) is a finite p– group. The first two claims are obvious and the third follows from [7; A. 4]. Since the normalizer N_{C}(T_{G}) of T_{G} taken in C maps onto π_{0}(C) and since W_{G} is generated by elements of order 2, the group C is connected. Because C and G have identical Weyl groups, they are isomorphic. This shows that P ⊂ Z(G). □

3.3 Remark. The only simply connected W –lattices, which are not polynomial, come from the Weyl group action of some exeptional connected compact Lie groups. Hence, for odd primes, the first and the third part of Proposition 3.2 are true for general W –lattices of a given pseudo reflection group W.

References.

[1] J.F. Adams & C.W. Wilkerson, Finite H–spaces and algebras over the Steenrod algebra, Ann. Math. 111 (1980), 95-143.

[2] J. Aguadé, Constructing modular classifying spaces, Israel J. Math. 66 (1989), 23-40.

[3] C.Broto & A. Viruel, Homotopy uniqueness of BP U(3), Preprint.

[4] C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.

[5] A. Clark & J. Ewing, The realization of polynomial algebras as cohomology rings, Pacific J. Math. 50 (1974), 425-434.

[6] W.G. Dwyer, H.Miller, & C.W. Wilkerson, The homotopical uniqueness of classifying spaces, Topology 31 (1992), 29-45.

[7] S. Jackowski, J. McClure & R. Oliver, Homotopy classification of self maps of BG via G–actions I, II, Ann. Math. 135 (1992), 183-226, 227-270.

[8] D. Notbohm, p-adic lattices of pseudo reflection groups, in: Proceedings of the Algebraic topology conference, Barcelona 1994, Ed.:C. Broto & all., Birkhäuser 1996.

[10] D. Notbohm, Topological realization of a family of pseudo reflection groups, preprint 1996.

[9] D. Notbohm, Spaces with polynomial cohomology, in preparation.

[11] G.C. Shepard and J.A. Todd, Finite unitary reflection groups, Can. J. Math. 6 (1957), 274-304.

[12] L. Smith, Polynomial invariants of finite groups, A.K. Peters Welley Massachusetts 1995.

[13] N.E. Steenrod, Polynomial algebras over the algebra of cohomology operations, Proceedings, Neuchâtel 1970, SLNM 196, Springer Verlag, 1971.

Mathematsisches Institut, Bunssenstr. 3-5, 37073 Göttingen, Germany. e-mail : notbohm at cfgauss.uni-math.gwdg.de