TOWARDS SPETSES I

Michel Broué, Gunter Malle, Jean Michel

January 18, 1999

To Claude Chevalley

Abstract.

We present a formalization, using data uniquely defined at the level of the Weyl group, of the construction and combinatorial properties of unipotent character sheaves and unipotent characters for reductive algebraic groups over an algebraic closure of a finite field. This formalization extends to the case where the Weyl group is replaced by a complex reflection group, and in many cases we get families of unipotent characters for a mysterious object, a kind of reductive algebraic group with a non real Weyl group, the “spets”.

In this first part, we present the general results about complex reflection groups, their associated braid groups and Hecke algebras, which will be needed later on for properties of “spetses”. Not all irreducible complex reflection groups will give rise to a spets (the ones which do so are called “spetsial”), but all of them afford properties which already allow us to generalize many of the notions attached to Weyl groups through the approach of “generic groups” (see [BMM1]).

Summary

Introduction

1. Complex reflection groups and generic Hecke algebras 1A. Generalities, degrees and codegrees 1B. Reflection groups and braid groups 1C. Generic Hecke algebras

2. Generic Hecke algebras as symmetric algebras 2A. The canonical trace form 2B. Schur elements, Generic degrees

3. Reflection data : first definitions 3A. Definitions 3B. Generic groups and reflection data 3C. The order of a reflection datum 3D. Various constructions with reflection data 3E. Classification of reflection data

4. Uniform class functions on a reflection datum 4A. Generalities, induction and restriction 4B. Degrees

5. Φ–Sylow theory and Φ-split Levi sub-reflection data 5A. The Sylow theorems

1991 Mathematics Subject Classification. Primary 20, 20G. We thank Meinolf Geck, George Lusztig and Raphaël Rouquier for fruitful conversations.

5B. Regular elements and Sylow theorems

6. Spetsial and cyclotomic specializations of Hecke algebras 6A. Cyclotomic Hecke algebras 6B. Some definitions and notation about palindromicity 6C. On spetsial cyclotomic Hecke algebras 6D. Palindromicity, fake degrees, generic degrees 6E. On generic degrees of cyclotomic Hecke algebras

Appendix : Symmetric algebras

Introduction

Weyl groups are finite groups acting as a reflection group on a rational vector space. These rational reflection group appear as the “skeleton” of many algebraic objects, like algebraic groups, Hecke algebras, braid groups, etc.

By extension of base field, Weyl groups are in particular complex reflection groups. Our aim is to present some evidence, results and conjectures, about the existence of analogues of reductive algebraic groups over finite fields with “Weyl group” a complex reflection group.

The results of Lusztig show that the set of unipotent characters of a “finite reductive group” only depends on the Weyl group of the associated algebraic group together with the action of the Frobenius endomorphism on it. In the course of this classification, Lusztig (see [Lu2]) observed that similar sets can also be attached to those finite real reflection groups which are not Weyl groups (namely, the finite Coxeter groups), just as if there existed a “fake algebraic group” whose Weyl group is non–crystallographic. It was at the conference held on the little Greek island named Spetses, during the summer of 1993, that we realized that a similar construction might also exist for non real reflection groups.

Since then, this was shown to be true ([Ma1], [BMM2]) for certain types of non– real irreducible reflection groups (called “spetsial”) by constructing sets of “unipotent characters”, Fourier transform matrices, and “eigenvalues of Frobenius”, which satisfy suitable generalizations of combinatorial properties of actual finite reductive groups. These were announced in the report [Ma4].

In this series of papers we present the appropriate general results on reflection groups and their associated Hecke algebras (which is the object of the present article), in order to apply them (which is the object of the subsequent article [BMM2]) to statement and description of properties of these strange creatures we call “Spetses”.

Some of our statements have a special status : they have only been proved for some of the irreducible complex reflection groups, but we conjecture they are true in general. We call such a statement a “Theorem–Assumption” (see 1.16, 2.1, 2.10) ; we state what are the known cases (for which we give proofs or references) and what are the left open cases. Assuming their validity in general allows us to deduce other general properties, and to state them without having to quote a list of possible exceptions – we only refer to the necessary “theorem–assumption”.

1. Complex reflection groups and generic Hecke algebras

1A. Generalities, degrees and codegrees.

For most of the results quoted here, we refer the reader to the classical literature on complex reflections groups, such as [Bou], [OrSo1], [OrSo2], [OrTe], [ShTo], [Sp]. As for general results about braid groups and generic Hecke algebras, we refer the reader to [BMR].

Notation.

The complex conjugate of a complex number λ is denoted by λ^{∗}. If P (t_{1}, t_{2}, …, t_{m}) is a Laurent polynomial in the set of indeterminates (t_{1}, t_{2}, …, t_{m}) with complex coefficients, P (t_{1}, t_{2}, …, t_{m})^{∗} denotes the Laurent polynomial whose coefficients are the complex conjugate of the coefficients of P (t_{1}, t_{2}, …, t_{m}).

For n a natural integer, we set ζ_{n} := exp(2πi/n) .

Let K be a subfield of the field of complex numbers ℂ. Let V be a K–vector space of dimension r. We denote by V^{∨} the dual space V^{∨} = Hom(V, K).

A pseudo–reflection of GL(V) is a non trivial element s of GL(V) which acts trivially on a hyperplane, called the reflecting hyperplane of s. Let W be a finite subgroup of GL(V) generated by pseudo–reflections. The pair (V, W) is called a K–reflection group.

A parabolic subgroup of W is by definition the subgroup of elements of W which act trivially on a subspace of V. The following result is due to Steinberg ([St1], Theorem 1.5) — cf. also exercises 7 and 8 in [Bou], Ch. v, §6.

1.1. Theorem. Let V′ be a subspace of V. Then the parabolic subgroup W_{V′}, consisting of all elements of W which fix V′ pointwise, is still generated by pseudo–reflections : W_{V′} is generated by those pseudo–reflections in W whose reflecting hyperplane contains V′.

We denote by A the set of reflecting hyperplanes of (V, W), and we set N := |A|. We denote by N^{∨} the number of pseudo-reflections in W (note that for real reflection groups we have N = N^{∨}).

For H ∈ A, we denote by W_{H} the pointwise stabilizer of H, and we set e_{H} := |W_{H}|. The group W_{H} is a minimal non trivial parabolic subgroup of W. All its non trivial elements are pseudo-reflections. The group W_{H} is cyclic : if s_{H} denotes the element of W_{H} with determinant ζ_{e_{H}} = exp(2iπ/e_{H}), we have W_{H} = ⟨s_{H}⟩, the group generated by s_{H}. Such a pseudo–reflection as s_{H} is called a distinguished pseudo–reflection in W.

The centralizer C_{W} (W_{H}) of W_{H} in W is also its normalizer, as well as the normalizer (set-wise stabilizer) of H.

For C ∈ A/W an orbit of hyperplanes, we denote by N_{C} its cardinality. We have N_{C} = |W : C_{W} (W_{H})| for H ∈ C. We also set e_{C} := e_{H} for H ∈ C.

We denote by SV the symmetric algebra of V, by SV^{W} the algebra of invariants of W, by SV_{+}^{W} the ideal of SV^{W} consisting of elements of positive degree. We set SV_{W} := SV/(SV_{+}^{W} .SV), and call it the coinvariant algebra of (V, W).

The following facts are known.

• Degrees.

There is a family of r integers d_{1}, d_{2}, …, d_{r} called the degrees of (V, W), defined as follows : the Poincaré polynomial of the graded module (V ⊗ SV_{W})^{W} is

x^{d_{1}−1} + x^{d_{2}−1} + ⋯ + x^{d_{r}−1} .

We have

(x + d_{1} − 1)(x + d_{2} − 1) ⋯ (x + d_{r} − 1) = ∑_{w∈W} x^{dimV^{⟨w⟩}}

(where V^{⟨w⟩} denotes the space of fixed points of w). It follows that

(1.2)  ∑_{j=1}^{j=r}(d_{j} − 1) = ∑_{H∈A}(e_{H} − 1) = ∑_{C∈A/W} N_{C}(e_{C} − 1) = N^{∨} .

• Codegrees.

There is a family of r integers d_{1}^{∨}, d_{2}^{∨}, …, d_{r}^{∨} called the codegrees of (V, W), defined by the following condition: the Poincaré polynomial of the graded module (V^{∨}⊗SV_{W})^{W} is

x^{d_{1}^{∨}+1} + x^{d_{2}^{∨}+1} + ⋯ + x^{d_{r}^{∨}+1} .

We have

(x − d_{1}^{∨} − 1)(x − d_{2}^{∨} − 1) ⋯ (x − d_{r}^{∨} − 1) = ∑_{w∈W} det_{V} (w)x^{dimV^{⟨w⟩}} .

It follows that

(1.3)  ∑_{j=1}^{j=r}(d_{j}^{∨} + 1) = ∑_{H∈A} 1 = ∑_{C∈A/W} N_{C} = N ,

and so

(1.4)  N + N^{∨} = ∑_{j=1}^{j=r}(d_{j} + d_{j}^{∨}) = ∑_{C∈A/W} N_{C}e_{C} .

Field of definition of characters of a reflection group.

The following theorem has been proved (using a case by case analysis) by Bessis [Bes] (see also [Ben]), and generalizes a well known result on Weyl groups.

1.5. Theorem–Definition.

Let (V, W) be a ℂ–reflection group. Let K be the field generated by the traces on V of all elements of W. Then all irreducible KW –representations are absolutely irreducible.

The field K is called the field of definition of the reflection group W.

We shall denote by μ(K) the group of roots of unity of the field K.

Coinvariant algebra and fake degrees.

The coinvariant algebra SV_{W}, viewed as a KW –module, is isomorphic to the regular representation of W (see [Ch], p. 779).

Moreover, SV_{W} is a graded W –module, and we denote

SV_{W} = ⨁_{n} SV_{W}^{n}

this graduation.

Let χ be a character of W. The fake degree of χ is the element of ℤ[x] defined as follows :

(1.6)  Feg_{χ}(x) := ∑_{n} ⟨tr(·, SV_{W}), χ⟩ x^{n} .

In particular, if χ is (absolutely) irreducible, its fake degree is the “graded” multiplicity of an irreducible representation with character χ in the graded module SV_{W}.

1.7. Example. Assume that W is a cyclic group of order e. The set of irreducible characters consists of the characters det_{V}^{j} for 0 ≤ j < e. Then the fake degree of det_{V}^{j} is x^{j}.

The following formulae are well known and they may be found, for example, in [Sp] (see also §3 below).

(1) We have

Feg_{χ}(x) = \frac{1}{|W |} ∑_{w∈W} \frac{χ(w)}{det(1 − xw)^{∗}} (∏_{j=1}^{j=r}(1 − x^{d_{j}})) .

(2) Since the coinvariant algebra SV_{W}, viewed as a KW –module, is isomorphic to the regular representation of W, we have

(1.8)  Feg_{χ}(1) = χ(1) .

The integer N(χ) is defined as

N(χ) := Feg′_{χ}(1) .

One writes sometimes

Feg_{χ}(x) = x^{e_{1}(χ)} + ⋯ + x^{e_{χ(1)}(χ)} ,

where the integers e_{j}(χ) are called the exponents of χ. Then we have

N(χ) = e_{1}(χ) + ⋯ + e_{χ(1)}(χ) .

The following result is a reformulation of a theorem of Gutkin (see [Gu]), for which we provide a new proof for the reader’s convenience.

1.9. Proposition. We have

N(χ) = ∑_{H∈A} N(Res_{W_{H}}^{W} χ) ,

where in the right hand side N is interpreted with respect to the reflection group (V, W_{H}).

Proof of 1.9. Let us consider the element S_{χ}(x) ∈ K(x) defined by

S_{χ}(x) := \frac{1}{|W |} ∑_{w∈W} \frac{χ(w)}{det(1 − xw)^{∗}} = (∏_{j=1}^{j=r}(1 − x^{d_{j}}))^{−1} Feg_{χ}(x) .

We define ψ(χ) by writing the first terms of the expansion of S_{χ}(x) as a Laurent series in (1 − x) as follows :

S_{χ}(x) = \frac{χ(1)}{|W |} \frac{1}{(1 − x)^{r}} + \frac{ψ(χ)}{|W |} \frac{1}{(1 − x)^{r−1}} + ….

Let us compute ψ(χ) in two different ways.

• We have ψ(χ) = |W |\frac{d}{dx}((1 − x)^{r}S_{χ}(x))_{|_{x=1}} . Since

∏_{j=1}^{j=r} d_{j} = |W | and ∑_{j=1}^{j=r}(d_{j} − 1) = N^{∨} ,

an easy computation gives

ψ(χ) = \frac{N^{∨}}{2} − N(χ) .

• On the other hand, if Ref(W) denotes the set of all reflections in W, we have

ψ(χ) = ∑_{ρ∈Ref(W)} \frac{χ(ρ)}{1 − det(ρ)^{∗}} .

Thus we get

(1.10)  N(χ) = \frac{N^{∨}}{2} − ∑_{ρ∈Ref(W)} \frac{χ(ρ)}{1 − det(ρ)^{∗}} = ∑_{ρ∈Ref(W)} \frac{1 − det(ρ)^{∗} + 2χ(ρ)}{2(1 − det(ρ)^{∗})} .

Since Ref(W) is the disjoint union of the sets Ref(W_{H}) for H ∈ A, 1.9 results now from the preceding formula. □

Let us denote by m_{C,j}^{χ} the multiplicity of det_{V}^{j} as a constituent of Res_{W_{H}}^{W} χ (for H ∈ C). In other words, we have, for all integer k,

χ(s_{H}^{k}) = ∑_{j=0}^{j=e_{C}−1} m_{C,j}^{χ}det_{V}^{j} (s_{H}^{k}) .

We denote by χ^{∗} the complex conjugate of the character χ, which is then the character of the contragredient representation of a representation defining χ.

The following properties may be found in [BrMi], 4.1 and 4.2.

1.11. Proposition. Whenever χ ∈ Irr(W), we have

(1) N(χ) = ∑_{C∈A/W} ∑_{j=0}^{j=e_{C}−1} j N_{C} m_{C,j}^{χ} ,

(2) N(χ) + N(χ^{∗}) = ∑_{C∈A/W} ∑_{j=1}^{j=e_{C}−1} N_{C}e_{C}m_{C,j}^{χ} ,

(3) χ(1)(N + N^{∨}) − (N(χ) + N(χ^{∗})) = ∑_{C∈A/W} N_{C}e_{C}m_{C,0}^{χ} .

Sketch of proof of 1.11.

(1) By 1.9 above, it suffices to prove that

N (Res_{W_{H}}^{W} χ) = ∑_{j=0}^{j=e_{C}−1} j m_{C,j}^{χ} ,

thus that N(Res_{W_{H}}^{W} det_{V}^{j}) = j , which is an immediate consequence of 1.7 above. The last two assertions result from the first one and from the equality

m_{C,j}^{χ} = m_{C,e_{C}−j}^{χ^{∗}} .

□ 1B. Reflection groups and braid groups.

From now on we assume that (V, W) is a complex reflection group (i.e., a ℂ–reflection group).

We set M := V −⋃_{H∈A} H and we denote by p: M → M/W the canonical surjection. Note that the spaces M and M/W are path connected.

Let x_{0} ∈ M. We introduce the following notation for the fundamental groups:

P := π_{1}(M, x_{0}) and B := π_{1}(M/W, p(x_{0})) ,

and we call B and P respectively the braid group (at x_{0}) and the pure braid group (at x_{0}) associated to W.

The covering M → M/W is a non ramified Galois covering with Galois group W, by Steinberg’s theorem (see Theorem 1.1 above), hence the projection p induces a surjective map B ↠ W , σ ↦ \overline{σ}, an anti–morphism, as follows :

Let σ̃: [0, 1] → M be a path in M, such that σ̃(0) = x_{0}, which lifts σ. Then \overline{σ} is defined by the equality \overline{σ}(x_{0}) = σ̃(1) .

Denoting by W^{op} the group opposite to W, we have the following short exact sequence :

(1.12)  1 → P → B → W^{op} → 1 ,

where the map B → W^{op} is defined by σ ↦ \overline{σ}.

We denote by β the path [0, 1] → M defined by

β: t ↦ x_{0} exp(2iπt/|Z(W)|) .

From now on, we assume that W acts irreducibly on V. It results then from Schur’s lemma that

Z(W) = {exp(2iπk/|Z(W)|) | (k ∈ ℤ)} ,

and so in particular β defines an element of B, which we will still denote by β. We denote by π the loop [0, 1] → M defined by

π: t ↦ x_{0} exp(2iπt) .

The following result may be found in [BMR]. It relies on the “diagram presentations” of B given in [BMR] (see loc. cit., 2.27).

1.13. Theorem. Assume W different from G_{24}, G_{27}, G_{29}, G_{31}, G_{33}, G_{34}.

The center Z(B) of B is infinite cyclic and generated by β, the center Z(P) of P is infinite cyclic and generated by π, and the short exact sequence (1.12) induces a short exact sequence

1 → Z(P) → Z(B) → Z(W) → 1 .

Remark. It is conjectured to be true for groups of type G_{24}, G_{27}, G_{29}, G_{31}, G_{33}, G_{34}.

Parabolic braid subgroups.

Let I be an intersection of reflecting hyperplanes : I = ⋂_{{H∈A \atop I⊆H}} H . We set

A_{I} := {H ∈ A | (I ⊆ H)} and M_{I} := V − ⋃_{H∈A_{I}} H .

We recall (see 1.1 above) that the parabolic subgroup W_{I}, the pointwise stabilizer of I, is generated by all the cyclic reflection groups W_{H} for H ∈ A_{I}.

For x_{0} ∈ M, we set

P_{I} := π_{1}(M_{I}, x_{0}) and B_{I} := π_{1}(M_{I}/W_{I}, p(x_{0})) .

We recall (see [BMR], Proposition 2.29) that there is a morphism of short exact sequences (see 1.12 above)

\xymatrix{1 \ar[r] & P_{I} \ar@{^{(}->}[d] \ar[r] & B_{I} \ar@{^{(}->}[d] \ar[r] & W_{I}^{op} \ar@{^{(}->}[d] \ar[r] & 1 \\ 1 \ar[r] & P \ar[r] & B \ar[r] & W^{op} \ar[r] & 1}

where all vertical arrows are injective, and where the corresponding injection B_{I} ↪ B is well–defined up to P –conjugation.

Any subgroup of B image of B_{I} by one of the preceding injections is called a parabolic braid subgroup of B associated with I. For a given I, such parabolic braid subgroups constitute a conjugacy class of subgroups under P.

Length functions.

• There is a unique length function ℓ: B → ℤ defined as follows (see [BMR], Prop. 2.19) : if b = s_{1}^{n_{1}} · s_{2}^{n_{2}} ⋯ s_{m}^{n_{m}} where (for all j) n_{j} ∈ ℤ and s_{j} is a generator of the monodromy around an element of A in B, then

ℓ(b) = n_{1} + n_{2} + ⋯ + n_{m} .

• More generally, given C ∈ A/W, there is a unique length function ℓ_{C}: B → ℤ (this is the function denoted by π_{1}(δ_{C}) in [BMR], see Prop. 2.16 in loc. cit.) defined as follows : if b = s_{1}^{n_{1}} · s_{2}^{n_{2}} ⋯ s_{m}^{n_{m}} where (for all j) n_{j} ∈ ℤ and s_{j} is a generator of the monodromy around an element of C_{j}, then

ℓ_{C}(b) = ∑_{{j | (C_{j}=C)}} n_{j} .

Thus we have, for all b ∈ B,

ℓ(b) = ∑_{C∈A/W} ℓ_{C}(b) .

• Notice that, by [BMR1], 2.25, we have

(1.14)  ℓ_{C}(π) = N_{C}e_{C} and ℓ(π) = N + N^{∨} .

Remark : Artin–like and Coxeter–like presentations.

• We say that B has an Artin–like presentation (cf. [Op2], 5.2) if it has a presentation of the form

⟨s ∈ S | {v_{i} = w_{i}}_{i∈I}⟩

where S is a finite set of generators of the monodromy, and I is a finite set of relations which are multi–homogeneous, i.e., such that (for each i) v_{i} and w_{i} are positive words in elements of S and (for each C) ℓ_{C}(v_{i}) = ℓ_{C}(w_{i}).

• Note that if B has an Artin–like presentation, it results from [BMR], 2.18(2) that W has the corresponding Coxeter–like presentation, namely

⟨s ∈ S | {v_{i} = w_{i}}_{i∈I}, {s^{e_{Cs}} = 1}_{s∈S}⟩

where C_{s} is such that s is a generator of the monodromy around an hyperplane of C_{s}.

It is proved in [BrMaRo] through a case–by–case analysis that, whenever W is an irreducible complex reflection group of type different from G_{24}, G_{27}, G_{29}, G_{31}, G_{33}, G_{34}, then its associated braid group has an Artin–like presentation.

Let us now prove that B has always an “almost Artin–like” presentation, i.e., a presentation as defined above except that the sets S and I need not be finite.

Indeed, by [BMR], 2.2, we know that B is generated by generators of the monodromy. Thus B has a presentation of the form

⟨s ∈ S | {v_{i} = w_{i}}_{i∈I}⟩

where v_{i} and w_{i} are words in the elements of S and their inverses. Let us show that, possibly adding a finite number of new generators and relations for each i ∈ I, we may assume that v_{i} and w_{i} are positive words. Since ℓ_{C}(v_{i}) = ℓ_{C}(w_{i}) whenever v_{i} and w_{i} define equal elements of B, we will then get the desired presentation (note that by [BMR], 2.8, we may chose S to be finite to start with ; however we also need to start with a finite I in order to end up with a finite S in our process).

We get rid of inverses of generators of S in the relation v_{i} = w_{i} by induction on the number of inverses appearing there. The induction step is as follows : assume for example that s^{−1} appears in v_{i} and that it is the leftmost occurrence of an inverse in v_{i}. Write v_{i} = as^{−1}b where a is a positive word. We then add a new generator of the monodromy s′ = asa^{−1} to S, and a new relation where both sides are positive words : as′ = sa, and replace the relation v_{i} = w_{i} by ab = s′w_{i} where one less inverse appears.

1C. Generic Hecke algebras.

Let

u = (u_{C,j})_{(C∈A/W)(0≤j≤e_{C}−1)}

be a set of ∑_{C∈A/W} (e_{C}) indeterminates. We set u^{−1} = (u_{C,j}^{−1})_{(C∈A/W)(0≤j≤e_{C}−1)} . We denote by ℤ[u, u^{−1}] the ring of Laurent polynomials in the indeterminates u.

Let ℑ be the two-sided ideal of the group algebra ℤ[u, u^{−1}]B generated by all the elements

(s_{H} − u_{C,0})(s_{H} − u_{C,1}) ⋯ (s_{H} − u_{C,e_{C}−1})

where C ∈ A/W, H ∈ C, and s_{H} runs over the set of generators of the monodromy around H in B.

Notice that, since all the generators of the monodromy around any hyperplane H ∈ C are conjugate under B (cf. for example [BMR], §2.B), it suffices to take one of the s_{H} in the above relations to generate the ideal ℑ.

1.15. Definition.

The generic Hecke algebra H(W, u) is the ℤ[u, u^{−1}]-algebra ℤ[u, u^{−1}]B/ℑ.

Note that the specialization u_{C,j} ↦ exp (2iπj/e_{C}) takes the generic Hecke algebra H(W, u) to the group algebra of W^{op} on a cyclotomic extension of ℤ.

Any specialization of the generic Hecke algebra H(W, u) through which the previous one factorizes is called admissible.

We shall denote by h: ℤ[u, u^{−1}]B → H(W, u) the natural epimorphism.

For t a ℤ[u, u^{−1}]–linear map defined on H(W, u), we shall often omit the letter h by writing t(b) instead of t(h(b)).

Grading the Hecke algebra.

The algebra ℤ[u, u^{−1}] is multi–graded over ℤ : for each C ∈ A/W, we define the grading

deg_{C} ℤ[u, u^{−1}] → ℤ

by setting

deg_{C}(u_{C′,j}) := \{ \begin{matrix}1 if C′ = C , \\ 0 if C′ ≠ C . \end{matrix}

If we set deg_{C}(b) := ℓ_{C}(b) for all b ∈ B and C ∈ A/W, we see then that the group algebra ℤ[u, u^{−1}]B is endowed with a grading on ℤ^{A/W}. Since the ideal ℑ (see above 1.15) is homogeneous for this grading, it follows that the maps deg_{C} endow the generic Hecke algebra H(W, u) with a structure of graded algebra over ℤ^{A/W}.

Parabolic Hecke sub–algebras.

Let I be an intersection of reflecting hyperplanes, and let W_{I} (the pointwise stabilizer of I) be the corresponding parabolic subgroup of W (see above). We recall that we denote by A_{I} the set of reflecting hyperplanes of W_{I}.

Let u′ = (u′_{C,j})_{(C∈A_{I}/W_{I})(0≤j≤e_{C}−1)} be a set of ∑_{C∈A_{I}/W_{I}} (e_{C}) indeterminates.

Let a : A_{I}/W_{I} → A/W be the application which maps a W_{I}-orbit of hyperplanes to the corresponding W -orbit.

We have a morphism

ℤ[u′, u′^{−1}] → ℤ[u, u^{−1}] , u′_{C,j} ↦ u_{a(C),j} .

The injection B_{W_{I}} ↪ B_{W}, defined up to P –conjugation, (see above) induces an inclusion, defined up to P –conjugation (see [BMR], §4)

H(W_{I}, u′) ⊗_{ℤ[u′,u′^{−1}]} ℤ[u, u^{−1}] → H(W, u) ,

whose image is called the generic parabolic Hecke sub–algebra of H(W, u) associated with I, and is denoted by H(W_{I}, W, u) (note that only the P –conjugacy class of this subalgebra is well defined).

Semi–simple Hecke algebras and absolutely irreducible characters.

The following assertion is known to hold

• for all infinite families of irreducible reflection groups by [ArKo], [BrMa2], [Ar] (see proof below),

• for all Coxeter groups (see for example [Bou], chap. IV, §2, ex. 23),

• for all groups of rank 2 (see [BrMa2] and [Ma2]),

• and for the group G_{25} (see [BrMa2]).

Thus it remains to be checked for the 8 groups G_{24}, G_{26}, G_{27}, G_{29}, G_{31}, G_{32}, G_{33}, G_{34}. We conjecture it to be true in all cases (see [BMR], §4).

It shows in particular that the algebra H(W, u) is a free graded ℤ[u, u^{−1}]–module of rank |W |, since it has a basis over ℤ[u, u^{−1}] consisting of homogeneous elements.

1.16. Theorem–Assumption 1. There exists a family (b_{w})_{w∈W} of |W | elements of the braid group B with

• b_{w} maps to w through the natural morphism B ↠ W^{op},

• b_{1} = 1, such that the family (h(b_{w}))_{w∈W} is a basis of H(W, u) over ℤ[u, u^{−1}].

Remark. It follows from [BMR], §4 (see loc. cit., 4.24) that it is enough to check that a family (h(b_{w}))_{w∈W} generates H(W, u) over ℤ[u, u^{−1}] to ensure that it is a basis.

Proof of 1.16 for the infinite series.

Let us first consider the case where W = G(d, 1, r). Then it follows from [BreMa] that if for any element w ∈ W we choose a reduced expression of w in a standard set of generators of W and if we denote by b_{w} the preimage of w in B with respect to this reduced expression, then the family (h(b_{w}))_{w∈W} is a basis of H(W, u) over ℤ[u, u^{−1}].

Now let us set W := G(de, e, r). It is well known (see for example [Ma1], Def. 5.9) that the generic algebra H(W, u) embeds as a subalgebra in an admissible specialization H_{e}(de, 1, r; u) of the generic algebra of G(de, 1, r), in such a way as to fulfill the following property : let us still denote by h the morphism from the braid group B(de, 1, r) into H_{e}(de, 1, r; u) ; then the family (h(b_{w}))_{w∈G(de,1,r)} is a basis of H_{e}(de, 1, r; u), and the subset (h(b_{w}))_{w∈W} is contained in the subalgebra H(W, u).

The family (h(b_{w}))_{w∈W} generates a free submodule of H(W, u) with the same rank, and this submodule is a direct summand in the larger free module H_{e}(de, 1, r; u). This implies that the family (h(b_{w}))_{w∈W} is indeed a basis of H(W, u). □

By Appendix, 7.4, (1) below, it follows that

1.17. Proposition. The ℚ(u)–algebra ℚ(u)H(W, u) is trace symmetric, hence separable and in particular semi–simple.

Let ℚ(μ_{∞}) be the subfield of ℂ generated by all roots of unity. Assume that (V, W) is an irreducible K–reflection group, where K is a subfield of ℚ(μ_{∞}) of finite degree over ℚ. We denote by μ(K) the group of all roots of unity in K. We denote by K(u) the field of fractions of the Laurent polynomial ring K[u, u^{−1}], where as above u = (u_{C,j})_{(C∈A/W)(0≤j≤e_{C}−1)} .

The following theorem is proved for any reflection group in [Ma3] (see theorem 5.2. in loc. cit.), provided that the corresponding Hecke algebra is defined by generators and relations symbolized by the diagrams of [BMR]. Since this last fact is unknown for the six exceptional groups G_{24}, G_{27}, G_{29}, G_{31}, G_{33}, G_{34}, we must call it a “theorem– assumption” : it is proved for all groups but the six previous ones.

1.18. Theorem–Assumption. We assume now that assumption 1 holds. Let t = (t_{C,j})_{(C∈A/W)(0≤j≤e_{C}−1)} be a set of ∑_{C∈A/W} e_{C} indeterminates such that,

for all C, j, we have t_{C,j}^{|μ(K)|} = ζ_{e_{C}}^{−j}u_{C,j} . The field K(t) is a splitting field for the Hecke algebra ℚ(u)H(W, u).

The field K(t) as above is called in what follows a good splitting field.

Let us denote by ℤ_{K} the ring of integers of K.

By Appendix, 7.2 below, the specialization

t_{C,j} ↦ 1

induces a bijection χ ↦ χ_{t} from the set Irr(W^{op}) of absolutely irreducible characters of W^{op} onto the set Irr(H(W, u)) of absolutely irreducible characters of the Hecke algebra H(W, u), such that the following diagram is commutative

H(W, u) ⟶^{χ_{t}} ℤ_{K}[t, t^{−1}]
↓  ↓
ℤ_{K}W^{op} ⟶^{χ} ℤ_{K} .

1.19. Convention. As functions on the underlying set of W, the characters of W and of W^{op} coincide. We identify Irr(W) and Irr(W^{op}). Thus we get a bijection

Irr(W) ⟶^{∼} Irr(H(W, u)) , χ ↦ χ_{t}

such that, for b ∈ B with image \overline{b} ∈ W, we have

(χ_{t}(b))_{|_{t=1}} = χ(\overline{b}) .

Linear characters.

Let Hom(B, ℚ(u)^{×}) denote the group of linear characters of B with values in ℚ(u)^{×}. We have an isomorphism (see for example [BMR], prop. 2.16) :

Hom(B, ℚ(u)^{×}) ≃ \underbrace{ℚ(u)^{×} × ℚ(u)^{×} × ⋯ × ℚ(u)^{×}}_{|A/W | times} ,

given by the map

θ ↦ (θ(s_{C}))_{C∈A/W} ,

where we denote by s_{C} a generator of the monodromy around a hyperplane which belongs to C.

It follows from what precedes that the linear characters of H(W, u) are induced by those linear characters of B described as follows : there exists a family of integers j := (j_{C})_{C∈A/W} where j_{C} ∈ {0, 1, …, e_{C} − 1}, such that s ↦ u_{C,j_{C}} if s is a generator of the monodromy around an element of C. We denote by θ_{j} this linear character of H(W, u).

Recall that we denote by det_{C} the linear character of W such that

det_{C}(s_{H}) = \{ \begin{matrix}det_{V} (s_{H}) if H ∈ C , \\ 1 if H ∉ C , \end{matrix}

where s_{H} is any reflection with reflecting hyperplane H.

Then the character (det_{C,j})_{t} of the Hecke algebra corresponding to det_{C}^{j} is defined by

(det_{C,j})_{t}(s_{H}) = \{ \begin{matrix}u_{C,j} if H ∈ C , \\ u_{C′,0} if H ∈ C′ , C′ ≠ C , \end{matrix}

where s_{H} is a generator of the monodromy around H.

We see in particular that (det_{C,j})_{t} is rational over ℚ(u) : we have

(1.20)  (det_{C,j})_{t} : H(W, u)^{×} → ℤ[u, u^{−1}]^{×} .

Notice that (with notation introduced above), for j = (j_{C})_{C∈A/W} we have

θ_{j} = ∏_{C∈A/W} (det_{C,j_{C}})_{t} .

Central morphisms associated with irreducible characters.

More generally, let χ ∈ Irr(W). Recall that, for C ∈ A/W, we denote by m_{C,j}^{χ} the multiplicity of det_{V}^{j} in the restriction of χ to the cyclic group W_{H}.

It results from [BrMi], 4.17, that

\frac{m_{C,j}^{χ}N_{C}e_{C}}{χ(1)} ∈ ℕ .

Since K(t) is a splitting field for the algebra K(t)H(W, u), the irreducible character χ_{t} defines an algebra morphism from the center of K(t)H(W, u) onto K(t) which we denote by

ω_{χ_{t}} : Z(K(t)H(W, u)) → K(t) .

It results from [BrMi], prop. 4.16, that

ω_{χ_{t}}(π) = exp(−2iπN(χ)/χ(1)) ∏_{A/W} ∏_{j=0}^{j=e_{C}−1} uC,j^{\frac{m{χ \atop C,j} N_{C} e_{C}}{χ(1)}} .

Now using the equalities t_{C,j}^{|μ(K)|} = exp(−2iπj/e_{C})u_{C,j} and 1.11 above, we get the equivalent formula

(1.21)  ω_{χ_{t}}(π) = ∏_{C∈A/W} ∏_{j=0}^{j=e_{C}−1} t_{C,j}^{|μ(K)|^{\frac{m{χ \atop C,j} N_{C} e_{C}}{χ(1)}}} .

A proof like in [BrMi], 4.16 provides the following generalization of formula 1.21.

1.22. Proposition. Assume that w ∈ B is such that w^{d} = π (for some natural integer d). Let w denote its image in W. Then we have

χ_{t}(w) = χ(w) ∏_{C∈A/W} ∏_{j=0}^{j=e_{C}−1} tC,j^{\frac{|μ(K)|}{d} \frac{m{χ \atop C,j} N_{C} e_{C}}{χ(1)}} .

Characters as homogeneous functions.

1.23. Proposition. Assume assumption 1 holds. For all χ ∈ Irr(W) and b ∈ B, the value χ_{t}(b) (as an element of ℤ_{K}[t, t^{−1}]) is multi–homogeneous with degree |μ(K)|ℓ_{C}(b) in the indeterminates (t_{C,j})_{0≤j<e_{C}} (for all C ∈ A/W).

Proof of 1.23. This is an immediate application of Appendix, 7.1, (2) below. □

Galois operations.

Both extensions K(t)/K(u) and K(t)/ℚ(t) are Galois (although in general the extension K(t)/ℚ(u) is not Galois). Since H(W, u) is defined over ℚ(u), both groups Gal(K(t)/K(u)) and Gal(K(t)/ℚ(t)) act on the set of irreducible characters of the algebra K(t)H(W, u), hence, through the preceding bijection, both groups act on Irr(W).

• For g ∈ Gal(K(t)/K(u)) and χ ∈ Irr(W), we denote by g(χ) the irreducible character of W defined by the condition

g(χ_{t}) = (g(χ))_{t} .

• The group Gal(K(t)/ℚ(t)) is isomorphic (through the restriction map from K(t) down to K) to the Galois group Gal(K/ℚ). Through this isomorphism we get an action of Gal(K/ℚ) on Irr(W). It is easy to see that this action coincides with the usual action of Gal(K/ℚ) on Irr(W). For g ∈ Gal(K/ℚ) and χ ∈ Irr(W), we still denote by g(χ) the image of χ under g.

1.24. Proposition.

(1) The linear characters of W are fixed under the operations of Gal(K(t)/K(u)).

(2) Whenever χ ∈ Irr(W) and g ∈ Gal(K(t)/K(u)), we have

(a) χ(1) = g(χ)(1),

(b) χ(s) = g(χ)(s) whenever s is a reflection in W,

(c) N(χ) = N(g(χ)).

Proof of 1.24. We have seen above that the characters (det_{C,j})_{t} are rational over K(u), which proves (1). Let us prove (2).

By definition of the operation of Gal(K(t)/K(u)), if w is an element of B with image w in W, and if χ_{t}(w) ∈ K(u), then χ(w) = g(χ)(w) . Thus the assertion (a) is immediate, and to prove (b) it is enough to prove that, whenever s_{H} is a generator of the monodromy around H, then χ_{t}(s_{H}^{k}) ∈ K(u) for 0 ≤ k < e_{H}.

The absolutely irreducible characters of the parabolic subalgebra generated by h(s_{H}) are the characters (det_{H,j})_{u} defined as the restrictions of the characters (det_{C,j})_{t}, hence they are rational on K(u), which establishes assertion (b).

Now assertion (c) is an immediate consequence of the formula 1.10 above. □

Some anti-automorphisms of the generic Hecke algebra.

We denote by α ↦ α^{∨} the automorphism of ℤ[u, u^{−1}] consisting in the simultaneous inversion of the indeterminates.

1.25. Proposition. There is a unique anti-automorphism of H(W, u), denoted by a_{1}, such that :

(a1) For b ∈ B, we have

a_{1}(h(b)) = h(b^{−1}) .

(a2) For α ∈ ℤ[u, u^{−1}], we have

a_{1}(α) = α^{∨} .

Proof of 1.25. By definition of H(W, u), to define a ℤ–automorphism a of H(W, u), it suffices to define the images of the elements h(b) for b ∈ B, and to check that the ideal ℑ is stable under a. If we impose a(h(b)) = h(b^{−1}), we see that

a((s_{H} − u_{C,0})(s_{H} − u_{C,1}) ⋯ (s_{H} − u_{C,e_{C}−1})) =

(−1)^{e_{C}}a( ∏_{j=0}^{j=e_{C}−1} u_{C,j})s_{H}^{−e_{C}} (s_{H} − a(u_{C,0})^{−1}) ⋯ (s_{H} − a(u_{C,e_{C}−1})^{−1}) .

In particular, if a satisfies (a2), we see that

\begin{matrix}a( & (s_{H} − u_{C,0})(s_{H} − u_{C,1}) ⋯ (s_{H} − u_{C,e_{C}−1})) = \\ (−1)^{e_{C}} ( ∏_{j=0}^{j=e_{C}−1} u_{C,j})^{−1} s_{H}^{−e_{C}} (s_{H} − u_{C,0}) ⋯ (s_{H} − u_{C,e_{C}−1}) . \end{matrix}

□

The group

𝔖_{W} := ∏_{C∈A/W} 𝔖_{e_{C}}

permutes the set of indeterminates u in an obvious way, thus acts on the ℤ–algebra ℤ[u, u^{−1}]. For σ ∈ 𝔖_{W}, we still denote by σ the corresponding automorphism of ℤ[u, u^{−1}], as well as the automorphism of ℤ–algebra of H(W, u) which extends the automorphism σ of ℤ[u, u^{−1}] by the condition σ(h(b)) = h(b) for all b ∈ B .

We define

(1.26)  a_{σ} := σ a_{1}

and 𝔖_{W} acts regularly on the set of all the anti-automorphisms a_{σ}.

Anti–automorphisms and characters of the generic Hecke algebra.

A linear character θ is transformed into another linear character by an anti-automor-phism a through the formula

aθ := aθa^{−1} .

It is immediate to check that, for σ ∈ 𝔖_{W}, we have a_{σ}θ_{j} = θ_{σ(j)} .

Notation.

• We extend the automorphism α ↦ α^{∨} of the ring ℤ[u, u^{−1}] to the automorphism (still denoted by α ↦ α^{∨}) of the ring ℤ_{K}[t, t^{−1}] such that

t_{C,j}^{∨} := t_{C,j}^{−1} (for all C and j), and λ^{∨} := λ^{∗} (for all λ ∈ ℤ_{K}) .

• We extend the anti–automorphism a_{1} to the algebra ℤ_{K}[t, t^{−1}]H(W, u) (hence to the algebra K(t)H(W, u)) by deciding that a_{1} induces the automorphism α ↦ α^{∨} on the ring ℤ_{K}[t, t^{−1}]. Thus, for λ ∈ K, b ∈ B, C ∈ A/W and 0 ≤ j < e_{C}, we have

a_{1}(λt_{C,j}h(b)) = λ^{∗}t_{C,j}^{−1}h(b^{−1}) .

• Whenever f: H(W, u) → ℤ_{K}[t, t^{−1}] is a function, we set (for h ∈ H)

(1.27)  f^{∨}(h) := f(a_{1}(h))^{∨} .

Thus, in particular, for b ∈ B we have f^{∨}(h(b)) := f(h(b^{−1}))^{∨} .

1.28. Lemma. For all χ ∈ Irr(W), we have χ_{t}^{∨} = χ_{t} .

Proof of 1.28. It suffices to check that χ_{t}^{∨} specializes to χ for the specialization t_{C,j} ↦ 1, which is clear. □

The opposite algebra.

The opposite algebra H(W, u)^{op} is defined as usual :

• as a ℤ[u, u^{−1}]–module, we have H(W, u)^{op} = H(W, u) ;

• the multiplication of two elements of H(W, u)^{op} is defined by (h, h′) ↦ h′h .

It is easy to check the following two properties.

1.29. The opposite algebra H(W, u)^{op} is

(a) isomorphic to ℤ[u, u^{−1}]B^{op}/ℑ^{op}, where ℑ^{op} denotes the ideal of the group algebra ℤ[u, u^{−1}]B^{op} generated by all the elements

(s_{H} − u_{C,0})(s_{H} − u_{C,1}) ⋯ (s_{H} − u_{C,e_{C}−1})

where C ∈ A/W, H ∈ C, s_{H} is a generator of the monodromy around H in B,

(b) semi–linearly isomorphic to the algebra H(W, u) through the map h ↦ h^{∨}.

2. Generic Hecke algebras as symmetric algebras

2A. The canonical trace form.

An element P (u) ∈ ℤ[u, u^{−1}] is called “multi–homogeneous” if, for each C ∈ A/W, it is homogeneous as a Laurent polynomial in the indeterminates u_{C,j} for j = 0, …, e_{C} −1 .

The following assertion is conjectured to be true for all reflection groups.

• We shall prove in general below that if there exists a form satisfying all conditions (1),(a), (b), (c) of 2.1, then it is unique.

• Theorem–Assumption 2 is proved in general for W a Coxeter group.

• For all infinite families of non Coxeter complex reflection groups, the existence of a form satisfying all properties stated in 2.1 below but property (1)(c) is proved without restriction. Property (1)(c) is a consequence of an additional assumption about “Schur elements” (see 2.9 below) which is due to G. Malle (see [Ma1]). Thus the unicity of the form t_{u} (which depends in particular on condition (1)(c)) still relies on 2.9.

• Finally, Theorem–Assumption 2 is only partly checked for the non Coxeter exceptional groups. A “good” candidate for t_{u} is known for all groups : it satisfies all properties below but property (1)(a). Property (1)(a) is probably easily checkable on computer for some of the small groups, but it is still open for large exceptional non Coxeter reflection groups.

2.1. Theorem–Assumption 2.

(0) Assumption 1 holds.

(1) There exists a unique linear form

t_{u} : H(W, u) → ℤ[u, u^{−1}]

with the following properties.

(a) t_{u} is a central form on the algebra H(W, u), i.e., for all h, h′ ∈ H(W, u), we have t_{u}(hh′) = t_{u}(h′h), and it endows H(W, u) with a structure of symmetric algebra on the ring ℤ[u, u^{−1}] i.e., the map

t̂_{u} : H(W, u) → Hom(H(W, u), ℤ[t, t^{−1}]) , h ↦ (h′ ↦ t_{u}(hh′))

is an isomorphism of ℤ[u, u^{−1}]–modules between H(W, u) and its ℤ[u, u^{−1}]–dual.

(b) Through the specialization u_{C,j} ↦ exp (2iπj/e_{C}) , the form t_{u} becomes the canonical linear form on the group algebra.

(c) For all b ∈ B, we have

t_{u}(b^{−1})^{∨} = \frac{t_{u}(bπ)}{t_{u}(π)} .

(2) The form t_{u} satisfies the following conditions.

(a) For b ∈ B, t_{u}(b) is invariant under the action of 𝔖_{W}.

(b) As an element of ℤ[u, u^{−1}], t_{u}(b) is multi–homogeneous with degree ℓ_{C}(b) in the indeterminates u_{C,j} for j = 0, 1, …, e_{C} − 1 . In particular, we have

t_{u}(1) = 1 and t_{u}(π) = (−1)^{N^{∨}} ∏_{{H∈A \atop 0≤j≤e_{H} −1}} u_{H,j} .

(c) If W′ is a parabolic subgroup of W, the restriction of t_{u} to a parabolic sub– algebra H(W′, W, u) is the corresponding specialization of t_{u′}(W′) (we recall that H(W′, W, u) is a specialization of H(W′, u′)).

What we can prove of 2.1.

We first give a general proof of the unicity of a form satisfying properties (1)(a), (b), (c). We shall then give a proof of its existence in the cases where it is known.

1. Proof of the unicity.

We shall prove a slightly more general result.

Let t = (t_{C,j})_{(C∈A/W)(0≤j≤e_{C}−1)} be a set of ∑_{C∈A/W} e_{C} indeterminates such that,

for all C, j, we have t_{C,j}^{|μ(K)|} = exp (−2iπj/e_{C}) u_{C,j} , (we recall μ(K) is the group of roots of unity of the field K), so that the field K(t) is a splitting field for the Hecke algebra K(u)H(W, u) (see 1.18 above).

2.2. Proposition. Assume that there exists a form t_{u} satisfying properties (1)(a), (b),

(c) of Theorem–Assumption 2 (2.1). Let t′ be a linear form

t′: K[t, t^{−1}]H(W, u) → K[t, t^{−1}]

such that (a) t′ is symmetric as a linear form on the algebra K[t, t^{−1}]H(W, u).

(b) Through the specialization t_{C,j} ↦ 1, t′ becomes the canonical form on the group algebra of W.

(c) For all h ∈ K[t, t^{−1}]H(W, u), we have

t′(a_{1}(h))^{∨} = \frac{t′(hπ)}{t′(π)} .

Then t′ = t_{u}.

Proof of 2.2.

• Let us first notice some simple consequences of properties (b) and (c).

Since π specializes onto 1, we see that t′(π) specializes to 1, hence in particular t′(π) ≠ 0, which gives sense to (c). Now apply (c) to h = 1 ; we get t′(1) = 1. Then by applying it to h = π^{−1}, we see that t′(π)t′(π)^{∨} = 1 , which shows that t′(π) is invertible in K[t, t^{−1}], hence is a monomial in t. Since t′(π) specializes to 1 when all t_{C,j} map to 1, we see then that t′(π) is a unitary monomial in t.

• Since t′ is symmetric, and since t_{u} is non degenerate, there exists an element z ∈ Z(K[t, t^{−1}]H(W, u)) (the center of K[t, t^{−1}]H(W, u)) such that

t′(h) = t_{u}(hz) for all h ∈ H(W, u) .

Indeed, the map t̂_{u} : K[t, t^{−1}]H(W, u) → Hom(H(W, u), K[t, t^{−1}]) is an isomorphism. One defines z as the pre–image of t′.

• Property (c) may then be written

t′(a_{1}(h))^{∨} = \frac{t_{u}(hπz)}{t_{u}(π)}.\frac{t_{u}(π)}{t′(π)} .

The left hand side of the previous equality is also equal to t_{u}(a_{1}(h)z)^{∨}, while the first term of the right hand side equals t_{u}(a_{1}(h)a_{1}(z))^{∨}, from which we deduce

t_{u}(a_{1}(h)z)^{∨} = \frac{t_{u}(π)}{t′(π)}t_{u}(a_{1}(h)a_{1}(z))^{∨} ,

hence that

a_{1}(z) = (\frac{t′(π)}{t_{u}(π)})^{∨} z

since the map t̂_{u}(W) was assumed to be an isomorphism.

Since a_{1} has order 2, it follows that

\frac{t′(π)}{t_{u}(π)} (\frac{t′(π)}{t_{u}(π)})^{∨} = 1 .

The element M(t) := \frac{t′(π)}{t_{u}(π)} is a unitary monomial in t. Hence we have

z = M(t)a_{1}(z) .

• We recall that the specialization t_{C,j} ↦ 1 induces a bijection denoted by χ ↦ χ_{t} from the set Irr(W) of absolutely irreducible characters of W onto the set Irr(H(W, u)) of absolutely irreducible characters of the Hecke algebra H(W, u).

Now let χ ∈ Irr(W) . We have

ω_{χ_{t}^{∨}}(z) = ω_{χ_{t}}(a_{1}(z))^{∨} .

Since χ_{t}^{∨} = χ_{t} (see 1.28 above), we see then that

ω_{χ_{t}}(z) = M(t)ω_{χ_{t}}(z)^{∨} .

• Since z is integral over K[t, t^{−1}], and since K[t, t^{−1}] is integrally closed in K(t), we see that ω_{χ_{t}}(z) ∈ K[t, t^{−1}]. Thus the previous equality implies that

ω_{χ_{t}}(z) = ε(χ)Ω(t)

where ε(χ) = ±1, and Ω(t) is the unitary monomial in t such that Ω(t)^{2} = M(t).

Let e_{χ_{t}} be the primitive idempotent of the split semi–simple algebra K(t)H(W, u) corresponding to the character χ_{t}. Then we have

z = ∑_{χ∈Irr(W)} ω_{χ_{t}}(z) e_{χ_{t}} ,

hence

z = Ω(t) ∑_{χ∈Irr(W)} ε(χ) e_{χ_{t}} .

By the specialization t_{C,j} ↦ 1 the forms t_{u} and t′ both go to the canonical form on the group algebra of W, hence the element z must go to 1. But the above formula shows that z specializes to ∑_{χ∈Irr(W)} ε(χ) e_{χ}. We deduce from this that, for all χ ∈ Irr(W), we have ε(χ) = +1, so z = Ω(t)1 and t′ = Ω(t)t_{u} . Applying the previous equality to π, we get t′(π) = Ω(t)t_{u}(π) .

Since we have (see above) Ω(t)^{2} = M(t), and (by definition of M(t)) t′(π) = M(t)t_{u}(π) , it follows that Ω(t) = M(t) = 1, hence t′ = t_{u}. □

2. Proof of the existence of the canonical form.

Let us first examine its existence in the case where W is a Coxeter group.

• In this case, A is the complexified of a real hyperplane arrangement. If S denotes the set of walls of a chamber of that real arrangement, {s_{H} | H ∈ S} is a Coxeter generating system for W. It is known (see for example [BMR], th. 2.26) that there is a set S := {s_{H} | H ∈ S}, where each s_{H} is a generator of the monodromy in B around H, such that S, together with the Coxeter diagram of (W, S), defines a presentation of B.

• We denote by B^{+} the monoid generated by {s_{H} | H ∈ S}. Then there is a unique map W ↪ B^{+} , w ↦ b_{w} , such that b_{s_{H}} = s_{H} and ℓ(b_{w}) equals the length of w on S.

• Let us define two characters of H(W, u) as follows. If s is a generator of the monodromy around a hyperplane H which belongs to the orbit C ∈ A/W, we set

\begin{matrix}θ_{0}(s) := u_{C,0} \\ θ_{1}(s) := u_{C,1} . \end{matrix}

2.3 Lemma.

(1) The family {h(b_{w})}_{w∈W} is a basis of H(W, u).

(2) We define the form t_{u} by the conditions

t_{u}(1) = 1 and t_{u}(h(b_{w})) = 0 for w ≠ 1 .

Then the following family is a dual basis with respect to t_{u} :

(\frac{(−1)^{ℓ(w)}h(b_{w^{−1}})}{θ_{0}(h(b_{w}))θ_{1}(h(b_{w}))})_{w∈W} .

Proof of 2.3.

The Iwahori–Hecke algebra H_{q} associated to W with parameters q := {q_{C}}_{C∈A/W} and generators {T_{s_{H}} }_{H∈S} satisfying (T_{s_{H}} −q_{C})(T_{s_{H}} +1) = 0 (where H ∈ C) is isomorphic to H(W, u) by setting T_{s_{H}} = −h(s_{H})/u_{C,1} and q_{C} = −u_{C,0}/u_{C,1}. The second assertion is just a translation by this isomorphism of the well–known fact that T_{b_{w}} and θ(b)^{−1}T_{b_{w^{−1}}} are dual bases of H_{q}, where T_{b} is the image of b under the map induced by s_{H} ↦ T_{s_{H}} , and where θ is the linear character defined by T_{s_{H}} ↦ q_{C}. □

The preceding lemma shows that t_{u} satisfies properties (1)(a), (1)(b) of 2.1.

Let us check that property (2)(c) of 2.1 also follows from the preceding lemma.

We keep the choices and notation (S, B^{+}, (b_{w})_{w∈W}) introduced before lemma 2.3. Up to conjugating by an element of the braid group B, we may assume that there exists a subset S′ of S such that the group B′ generated by S′ is a parabolic subgroup of B associated with W′. It follows that the family (b_{w})_{w∈W′} is a basis of the corresponding parabolic subalgebra H(W′, W, u) of H(W, u), and so that the restriction of t_{u} to this subalgebra is indeed the specialization of the corresponding form on the generic algebra of W′.

To check (1)(c), notice that it can be written t_{u}(a_{1}(b)) = t_{u}(bπ)/t_{u}(π). Since t_{u} is linear and a_{1} is an anti–automorphism, it is enough to check this property for a basis. We may use {h(b_{w}^{−1})}_{w∈W} as a basis (that this is a basis results from the fact that a_{1} is an anti–automorphism). We thus have to check that t_{u}(b_{w})^{∨} = t_{u}(b_{w}^{−1}π)/t_{u}(π). But b_{w}^{−1}π = (b_{w}^{−1}b_{w_{0}})b_{w_{0}}, and b_{w}^{−1}b_{w_{0}} is of the shape b_{w′} for some w′ ∈ W. From 2.3 it results that t_{u}(b_{w}^{−1}π) = 0 unless w = 1 where it is equal to t_{u}(π).

In order to check properties (2)(a) and (2)(b) of 2.1, we prove a more precise statement.

2.4. Lemma. Choose S a set of generators of B associated with the set of walls of a chamber of W (see remark above). Let b be an element of B^{+}, and let

h(b) = ∑_{w∈W} a_{b,w}h(b_{w})

be the decomposition of its image h(b) in the generic Hecke algebra H(W, u) on the usual basis (see 2.3 above). Then for each w ∈ W, the element a_{b,w} of ℤ[u, u^{−1}] is invariant under 𝔖_{W}, and homogeneous of degree ℓ(b) − ℓ(w).

Proof of 2.4. The proof is by induction on ℓ(b), and it relies on the following well known properties :

Let s ∈ S with image s in W and assume that (h(b_{s}) − u_{0})(h(b_{s}) − u_{1}) = 0 . Then for any w ∈ W, we have (with obvious notation)

h(s)h(b_{w}) = \{ \begin{matrix}h(b_{s}w) if ℓ(sw) = ℓ(w) + 1 , \\ (u_{0} + u_{1})h(b_{w}) − u_{0}u_{1}h(b_{sw}) if ℓ(sw) = ℓ(w) − 1 . \end{matrix}

□

Note that we have t_{u}(b) = a_{b,1} by 2.3 (2). □

Let us turn again to the general case where W is a complex reflection group.

For W a finite imprimitive irreducible complex reflection group :

• A form t_{u}, satisfying (1)(a), (1)(b) and (2)(c), is given in [BreMa] – the fact that t_{u} turns H_{u}(W) into a symmetric algebra is proved in [MM].

• We shall prove (1)(c), (2)(a) and (2)(b) in the next paragraph under an additional assumption (see 2.9 below).

Example. Let us examine the value of t_{u} in the case where W is cyclic (see [BrMa2]). We denote by e the order of W, we set u = {u_{0}, u_{1}, …, u_{e−1}}, and we denote by s the “positive” generator of B.

• If j > 0, t_{u}(s^{j}) is (−1)^{e−1} times the sum of all monomials in u, with degree j, where each indeterminate occurs with a strictly positive exponent.

• If j ≤ 0, t_{u}(s^{j}) is the sum of all monomials in u, with degree j (where each

indeterminate occurs with a non-positive exponent). Thus we have in particular

t_{u}(s^{j}) = 0 for 1 ≤ j < e ,

t_{u}(s^{e}) = (−1)^{e−1} u_{0}u_{1} ⋯ u_{e−1} ,

t_{u}(s^{−1}) = ∑_{j=0}^{j=e−1} u_{j}^{−1} .

Notice that, by (2)(c) in Theorem–Assumption 2, it results from what precedes that, for W any complex reflection group, we have

t_{u}(s) = 0 whenever s is a generator of the monodromy in B .

2B. Schur elements, Generic Degrees.

Schur elements.

2.5. Proposition. Assume that t_{u} is a linear form on H(W, u) such that assertion (1)(a) of 2.1 holds, i.e., the map

t̂_{u} : H(W, u) → Hom(H(W, u), ℤ[u, u^{−1}])

is an isomorphism.

(1) There exist elements S_{χ}(t) ∈ ℤ_{K}[t, t^{−1}] for χ ∈ Irr(W), such that

t_{u} = ∑_{χ∈Irr(W)} \frac{1}{S_{χ}(t)} χ_{t} .

(2) For all g ∈ Gal(K(t)/K(u)) we have S_{g(χ)}(t) = g(S_{χ}(t)) .

The element S_{χ}(t) is called the Schur element associated with χ.

Sketch of proof of 2.5.

The ring ℤ_{K}[t, t^{−1}] is integrally closed (see [Bou2], §3, Corollaire 3). Hence the existence of the Schur elements follows from Appendix, 7.12 below.

Assertion (2) follows from the fact that the functions χ_{t} (for χ ∈ Irr(W)) are linearly independent. □

It results from 1.20 that

(2.6)  S_{det_{C,j}}(t) ∈ ℤ[u, u^{−1}] .

2.7. Lemma. Assume that t_{u} is a linear form on H(W, u) such that the map

t̂_{u} : H(W, u) → Hom(H(W, u), ℤ[u, u^{−1}])

induces an isomorphism after extending scalars to K(t), and that t_{u}(1) = 1. Then the following assertions are equivalent :

(i) For all b ∈ B, we have

t_{u}^{∨}(b) = \frac{t_{u}(bπ)}{t_{u}(π)} .

(ii) There exists κ(t) ∈ K(t), κ(t) ≠ 0, such that, for all χ ∈ Irr(W), we have

κ(t)S_{χ}(t) = ω_{χ_{t}}(π)S_{χ}(t^{−1})^{∗} .

Proof of 2.7. On one hand, applying the involution f ↦ f^{∨} to the equality in 2.5 and using lemma 1.28, we get

t_{u}^{∨}(b) = ∑_{χ∈Irr(W)} \frac{1}{S_{χ}(t^{−1})^{∗}} χ_{t}(b) .

On the other hand, we have

t_{u}(bπ) = ∑_{χ∈Irr(W)} \frac{1}{S_{χ}(t)} χ_{t}(bπ)

= ∑_{χ∈Irr(W)} \frac{1}{S_{χ}(t)} ω_{χ_{t}}(π)χ_{t}(b) .

Using the fact that the functions χ_{t} (for χ ∈ Irr(W)) are linearly independent we can now prove 2.7.

(i) ⇒ (ii) : just take κ(t) = t_{u}(π). (ii) ⇒ (i) : We have, for all b ∈ B,

t_{u}^{∨}(b) = \frac{t_{u}(bπ)}{κ(t)} .

Since t_{u}(1) = 1, we see that κ(t) = t_{u}(π). □

The following palindromicity property of the Schur elements will be useful. It follows from the above proof.

2.8. Proposition. Assume assumption 2 holds. Whenever χ ∈ Irr(W), its Schur element S_{χ}(t) satisfies the following property :

S_{χ}(t^{−1})^{∗} = \frac{t_{u}(π)}{ω_{χ_{t}}(π)}S_{χ}(t) .

As announced before, we reconsider now the validity of Theorem–Assumption 2.

2.9. Assumption 3. For W an irreducible complex reflection group of the form G(d, 1, r), and for t_{u} as defined in [BreMa], the Schur elements are as conjectured in [Ma1], Vermutung 1.18.

It results from 2.7, assuming the values of Schur elements conjectured in 2.9 above and those computed in [Ma5], that assertion (1)(c) of 2.1 holds whenever assumption 1 holds.

Assertion (1)(a) of 2.1 has been proved in [MM], and assertion (2)(c) of 2.1 is immediate from the definition of t_{u} in [BreMa].

Assertions (2)(a) and (2)(b) of 2.1 result from the following Theorem–Assumption 4, where the basis from Assumption 1 and the symmetrizing form t_{u} from Theorem– Assumption 2 fit together.

As usual, we conjecture it to be true for all reflection groups. The current situation is as follows.

• Theorem–Assumption 4 is known for all Coxeter groups.

• It is proved for all infinite families of irreducible complex reflection groups (see proof below), for the form t_{u} studied in [MM].

• It is still open for most of (non Coxeter) exceptional groups. .

2.10. Theorem–Assumption 4. There exists a family (b_{w})_{w∈W} of |W | elements of the braid group B like in assumption 1, such that

t_{u}(h(b_{w})) = \{ \begin{matrix}1 if w = 1 , \\ 0 if w ≠ 1 . \end{matrix}

Proof of 2.10 for the infinite families.

It is shown in [MM] that the generic algebra H(G(d, 1, r), u) is symmetric over ℤ[u, u^{−1}] with respect to the form t_{u} which satisfies the properties of 2.10.

Let us set W := G(de, e, r). It is also shown in loc. cit. that the specialization of the canonical symmetrizing form of H(G(de, 1, r), u) to the algebra H_{e}(de, 1, r; u) (see proof of 1.16 above) restricts to a symmetrizing form for the generic algebra of H(W, u), which clearly satisfies the properties of 2.10. □

Absolute generic degrees and their palindromicity properties.

2.11. Definitions.

(1) The element P_{W} (u) ∈ ℤ[u, u^{−1}] is defined by

P_{W} (u) := S_{1}(t) .

(2) For χ ∈ Irr(W), the absolute generic degree of χ is the element of K(t) defined by

Deg_{χ}(t) := \frac{P_{W} (u)}{S_{χ}(t)} .

In particular, the absolute generic degree of the trivial character is 1.

Notice that, since t_{u} specializes at t_{C,j} = 1 (so u_{C,j} = ζ_{e_{C}}^{j}) onto the canonical form of the group algebra ℤW,

• P_{W} (u) specializes onto |W |,

• S_{χ}(t) specializes onto |W |/χ(1),

• Deg_{χ}(t) specializes onto χ(1).

Since

ω_{1_{t}}(π) = ∏_{A/W} u_{C,0}^{N_{C}e_{C}} and t_{u}(π) = (−1)^{N^{∨}} ∏_{A/W} ∏_{j=0}^{j=e_{C}−1} u_{C,j}^{N_{C}} ,

we get generalized “palindromicity properties” of the polynomial P_{W} (u) and of the absolute generic degrees.

We define the monomials M_{W} (u) and M_{χ}(u) as follows.

(2.12) \{ \begin{matrix}M_{W} (u) := \frac{ω_{1_{t}}(π)}{t_{u}(π)} = (−1)^{N^{∨}} ∏_{A/W} (u_{C,0}^{N_{C}(e_{C}−1)} ∏_{j=1}^{j=e_{C}−1} u_{C,j}^{−N_{C}} ) \\ M_{χ}(u) := \frac{ω_{1_{t}}(π)}{ω_{χ_{t}}(π)} = exp (2iπN(χ)/χ(1)) ∏_{A/W} (u_{C,0}^{N_{C}e_{C}} ∏_{j=0}^{j=e_{C}−1} u_{C,j}^{−^{\frac{m{χ \atop C,j} e_{C} N_{C}}{χ(1)}}} ) . \end{matrix}

We have

M_{W} (u) = \frac{S_{1}(t)}{S_{1}(t^{−1})^{∗}}

and

(2.13)  P_{W} (u) = M_{W} (u)P_{W} (u^{−1})^{∗} and Deg_{χ}(t) = M_{χ}(u)Deg_{χ}(t^{−1})^{∗} .

Remark. The polynomials S_{χ}(t), P_{W} (u), as well as the absolute generic degrees, depend by definition on the choice of the symmetrizing form t_{u}. Changing the choice of this form to t_{u}^{∨} would change

• S_{χ}(t) into S_{χ}(t^{−1})^{∗},

• P_{W} (u) into P_{W} (u^{−1})^{∗},
• Deg_{χ}(t) into Deg_{χ}(t^{−1})^{∗}.

3. Reflection data : first definitions

3A. Definitions.

From now on we assume K is a subfield of the field ℚ(μ_{∞}) generated by all roots of unity. We denote by z ↦ z^{∗} the complex conjugation, which induces an automorphism of K.

We denote by ℤ_{K} the ring of integers of K.

3.1. Definitions.

1. A reflection datum 𝔾 on K is a pair (V, W ϕ), where

• V is a finite dimensional vector space over K,

• W is a finite reflection group on V,

• ϕ is an element of finite order of GL(V) which normalizes W. Note that only the coset W ϕ (and not the automorphism ϕ) is defined by 𝔾.

The vector space V is called the space of the reflection datum and its dimension r is called the rank of the reflection datum. The group W is called the reflection group of the reflection datum. The image of W ϕ modulo W is denoted by \overline{ϕ} and called the twist of the reflection datum. Its order is denoted by δ(𝔾) : we have ϕ^{m} ∈ W if and only if δ(𝔾) divides m.

A reflection datum is called split if W ϕ = W, i.e., if ϕ ∈ W.

2. A sub–reflection datum of 𝔾 = (V, W ϕ) is a reflection datum of the shape 𝔾′ = (V′, W′(wϕ)_{|_{V ′}}), where V′ is a subspace of V, W′ is a reflection subgroup of N_{W} (V′)|_{V′} (the restriction to V′ of the stabilizer N_{W} (V′) of V′, isomorphic to N_{W} (V′)/C_{W} (V′)), and wϕ is an element of W ϕ which stabilizes V′ and normalizes W′.

3. A toric reflection datum (or torus) is a reflection datum whose reflection group is trivial. A torus of 𝔾 = (V, W ϕ) is a sub–reflection datum of the shape (V′, (wϕ)_{|_{V ′}}), where V′ is a subspace of V, and wϕ is an element of W ϕ which stabilizes V′.

4. If 𝕋 = (V′, (wϕ)_{|_{V ′}}) is a torus of 𝔾, the centralizer of 𝕋 in 𝔾 is the reflection datum defined by C_{𝔾}(𝕋) := (V, C_{W} (V′)wϕ) (notice that C_{W} (V′)wϕ is the set of all elements in W ϕ which act like wϕ on V′). Such a reflection datum is also called a Levi sub-reflection datum of 𝔾.

5. The center of a reflection datum 𝔾 = (V, W ϕ) is the torus Z(𝔾) := (V^{W} , ϕ_{|_{(V ^{W} )}}) (notice that it does not depend on the choice of ϕ in W ϕ).

6. We say that an extension K′ of K splits 𝔾 if it contains the m-th roots of unity, where m is the l.c.m. of the orders of the elements of W ϕ.

Remarks.

1. The reflection group of a reflection datum over ℚ (resp. over a subfield of ℝ) is a Weyl group (resp. a Coxeter group).

2. Any torus of 𝔾 is contained in a maximal torus of 𝔾, a reflection datum of the shape (V, wϕ) where w ∈ W.

3.2. Lemma.

(1) The Levi sub–reflection data of 𝔾 = (V, W ϕ) are the reflection data of the shape 𝕃 = (V, W′wϕ), where W′ is a parabolic subgroup of W and wϕ is an element of W ϕ which normalizes W′.

(2) If 𝕃 is a Levi sub–reflection datum of 𝔾, then C_{𝔾}(Z(𝕃)) = 𝕃.

Indeed, by the above definition of Levi sub–reflection datum (3.1, 4) it is enough to prove that if 𝕃 is defined as in (1), then C_{𝔾}(Z(𝕃)) = 𝕃. By 3.1, 5, we have Z(𝕃) = (V^{W′}, (wϕ)_{|_{(V ^{W′})}}), hence by 3.1, 3, we have C_{𝔾}(Z(𝕃)) = (V, C_{W} (V^{W′})wϕ), and the result follows from Steinberg’s theorem 1.1. □

Remark. The first assertion of lemma 3.2 above shows that the Levi sub–reflection data of 𝔾 are particular sub–reflection data of maximal rank of 𝔾.

3B. Generic groups and reflection data.

Now we relate the “generic groups” of [BMM1] to reflection data.

For the simplicity of the exposition, let us start by examining the case of generic groups over ℚ, before looking at the “very twisted” cases ^{2} B_{2}, ^{2} F_{4}, ^{2} G_{2}.

Let 𝔾 = (V, W ϕ) be a reflection datum on ℚ, so that W is a Weyl group. Let (R, R^{∨}) be a root system for W (so in particular R is a finite subset of V, while R^{∨} is a finite subset of V^{∨}). Let us denote by Q(R) and Q(R^{∨}) the ℤ-submodules of respectively V^{∨} and V generated respectively by R and R^{∨}.

We denote by V^{∨} := Hom(V, ℚ) the dual space of V, we still denote by W the image of W acting through the contragredient operation on the dual vector space V^{∨}, and we set ϕ^{∨} := ^{t}ϕ^{−1} .

• Any choice of a pair of dual lattices X and Y in respectively V^{∨} and V such that (a) X is W.ϕ^{∨}-stable and Y is W ϕ-stable,

(b) Q(R) ⊆ X and Q(R^{∨}) ⊆ Y provides a generic group :

𝔾_{(X,Y)} := ((X, R, Y, R^{∨}), W ϕ) .

• Reciprocally, any generic group ((X, R, Y, R^{∨}), W ϕ) defines a reflection datum

𝔾 := (ℚ ⊗_{ℤ} Y, W ϕ) ,

and we have 𝔾_{(X,Y)} = ((X, R, Y, R^{∨}), W ϕ).

Notice that two generic groups ((X, R, Y, R^{∨}), W ϕ) and ((X_{1}, R_{1}, Y_{1}, R_{1}^{∨}), W_{1}ϕ_{1}) such that ℚ ⊗ Y = ℚ ⊗ Y_{1} and W ϕ = W_{1}ϕ_{1} define the same reflection datum. In a sense, reflection data classify generic groups up to isogeny.

With the previous notation, the map 𝕃_{(X,Y)} ↦ 𝕃 is then a bijection between Levi sub– reflection data of 𝔾 and generic Levi subgroups of 𝔾_{(X,Y)} which respects all the invariants which will be introduced for reflection data and which were previously introduced (see [BMM1]) for generic groups (such as polynomial orders, signs, graded representation, class functions) or will be introduced later on (such as unipotent degrees). Moreover, it “commutes” with the usual constructions (adjoint, dual), and 𝕃 is Φ_{d}–split if and only if 𝕃_{(X,Y)} is.

Let us now turn our attention to the “very twisted cases”.

Let W be a Weyl group of type B_{2}, F_{4}, or G_{2}.

Let (R, R^{∨}) be a root system for W. As above, we denote by Q(R) and Q(R^{∨}) the free ℤ—modules generated respectively by R and R^{∨}.

Let p be 2 if W is of type B_{2} or F_{4}, and 3 if W is of type G_{2}.

Then there exists a finite order automorphism φ_{p} of ℤ[\sqrt{p}^{−1}] ⊗_{ℤ} Q(R) with the following properties.

• The automorphisms φ_{p}^{2} and \sqrt{p}φ_{p} both normalize Q(R) in ℤ[\sqrt{p}^{−1}] ⊗_{ℤ} Q(R).

• There exists a permutation σ of R such that

φ_{p}(σ(α)) = \sqrt{p}α,  φ_{p}^{∨}(α^{∨}) = (\sqrt{p})^{−1}(σ(α))^{∨} for α short,
φ_{p}(σ(α)) = (\sqrt{p})^{−1}α, φ_{p}^{∨}(α^{∨}) = \sqrt{p}(σ(α))^{∨}  for α long.

Indeed :

• Suppose W of type B_{2}. Let {α_{1}, α_{2}} a basis of R, where α_{1} is short and α_{2} is long. The other positive roots are α_{1} + α_{2} and 2α_{1} + α_{2}. Let σ be the involution which exchanges α_{1} and α_{2}, α_{1} + α_{2} and 2α_{1} + α_{2} (defined by symmetry on negative roots). We take φ_{2} to be the map defined by φ_{2}(α_{1}) = \sqrt{2}^{−1}α_{2} and φ_{2}(α_{2}) = \sqrt{2}α_{1}.

• Suppose W of type G_{2}. Let {α_{1}, α_{2}} a basis of R, where α_{1} is short and α_{2} is long. The other positive roots are α_{1} + α_{2}, 2α_{1} + α_{2}, 3α_{1} + α_{2}, 3α_{1} + 2α_{2}. Let σ be the involution which exchanges α_{1} and α_{2}, α_{1} + α_{2} and 3α_{1} + α_{2}, 2α_{1} + α_{2} and 3α_{1} + 2α_{2}, (defined by symmetry on negative roots). We take φ_{3} to be the map defined by φ_{3}(α_{1}) = \sqrt{3}^{−1}α_{2} and φ_{3}(α_{2}) = \sqrt{3}α_{1}.

Let K := ℚ(\sqrt{p}). Let V be a finite dimensional KW –module containing K⊗Q(R^{∨}) as a submodule, endowed with a finite order automorphism ϕ, which normalizes ℤ[\sqrt{p}^{−1}]⊗ Q(R^{∨}) and induces φ_{p}^{∨} on it.

• Any choice of a pair of dual free ℤ–modules X and Y with V^{∨} := K ⊗_{ℤ} X and V := K ⊗_{ℤ} Y such that

(a) Q(R) ⊆ X and Q(R^{∨}) ⊆ Y

(b) ℤ[\sqrt{p}^{−1}] ⊗ X is stable under W ϕ^{∨}, and ℤ[\sqrt{p}^{−1}] ⊗ Y is stable under W ϕ,

(c) X is stable under (ϕ^{∨})^{2} and under \sqrt{p}ϕ^{∨}, Y is stable under ϕ^{2} and under \sqrt{p}ϕ, provides what is called a “donnée radicielle complète” in [BrMa2] :

𝔾_{(X,Y)} := ((X, R, Y, R^{∨}), W ϕ) .

• Reciprocally, any “donnée radicielle complète” ((X, R, Y, R^{∨}), W ϕ) as above defines a reflection datum on K

𝔾 := (K ⊗_{ℤ} Y, W ϕ) ,

and we have 𝔾_{(X,Y)} = ((X, R, Y, R^{∨}), W ϕ).

Comment. The motivation for the definition of generic groups was to formalize the properties of reductive algebraic groups over a finite field which are independent of q. Indeed, to a generic group 𝔾_{(X,Y)} and a power q of a prime p, is associated an algebraic group G over \overline{F}_{q} and an isogeny F (associated to q and the map ϕ on V), well defined up to isomorphism (cf. [BMM1], 1.1). The properties of reflection data that we will discuss in the remaining of this paper are motivated by the properties of unipotent class functions on reductive groups (which have been observed to depend only on the associated reflection datum). We will reflect on that by comments of the form “for G this means” and “for G^{F} this means” where by G we will mean any of the algebraic groups over \overline{F}_{q} with reflection datum 𝔾 (in general, we will use the same letter as for the reflection datum, but in bold instead of blackboard bold font).

3C. The order of a reflection datum.

Let 𝔾 = (V, W ϕ) be a reflection datum. We set N^{∨}(𝔾) := N^{∨}(W) (or simply N^{∨}), the number of pseudo–reflections in W, and we set N(𝔾) := N(W) (or simply N), the number of reflecting hyperplanes of W.

Generalized degrees of a reflection datum.

Let K′ be an extension of K which splits 𝔾. We set V′ := K′ ⊗_{K} V.

Let SV be the symmetric algebra of V. Similarly, let SV′ be the symmetric algebra of V′. Since ϕ acts completely reducibly on the subspace of elements of (SV′)^{W} with given degree, we can always choose a family {P_{1}, …, P_{r}} of basic homogeneous invariants which are all eigenvectors for the action of ϕ (see [St2], 2.1), with eigenvalues respectively ζ_{1}, …, ζ_{r}. Then the family of pairs {(d_{1}, ζ_{1}), …, (d_{r}, ζ_{r})} depends only on 𝔾 (cf. for example [Sp], 6.1), and we call them the generalized degrees of the reflection datum 𝔾.

Generalized sign.

Let (SV)_{+}^{W} be the ideal of (SV)^{W} consisting of elements without degree zero terms. The vector space (SV)_{+}^{W} /((SV)_{+}^{W})^{2} has dimension r, and is endowed with an action of the image \overline{ϕ} of ϕ modulo W. Any family of basic homogeneous invariants \{P_{1}, …, P_{r}\} as above provides a basis of (SV′)_{+}^{W} /((SV′)_{+}^{W})^{2} on which \overline{ϕ} is diagonal. The generalized sign (note that in general it is not a sign !) of the reflection datum 𝔾 is by definition :

ε_{𝔾} := (−1)^{r}ζ_{1} ⋯ ζ_{r} .

One can prove (cf. [Sp], 6.5) that if there exists an element wϕ ∈ W ϕ such that wϕ admits a fixed point in V −⋃_{H∈A} H, then {ζ_{1}, …, ζ_{r}} is the spectrum of ϕ (in its action on V′). In particular, we have then ε_{𝔾} := (−1)^{r}det_{V} (ϕ).

It follows that

3.3. If 𝕋 = (V, wϕ) is a maximal torus of 𝔾, we have ε_{𝕋} = (−1)^{r}det_{V} (wϕ). Moreover, if there exists an element wϕ ∈ W ϕ such that wϕ admits a fixed point in V − ⋃_{H∈A} H, then ε_{𝕋} = ε_{𝔾}det_{V} (w).

Remark. Contrary to the case of real reflection data, the group ⟨W ϕ⟩ (subgroup of GL(V) generated by W ϕ) is not always a semi–direct product.

For example, for W = G(4, 2, 4), there exists an element ϕ of order 4 in N_{GL(V)}(W) such that ⟨W ϕ⟩ = G(4, 1, 4), which is a non split extension of W (see proposition 3.12 below).

The graded regular representation.

We set R𝔾 := SV/(SV.(SV)_{+}^{W}) (viewed as endowed with the natural action of the group W ⟨ϕ⟩). Then R𝔾 is a finite dimensional graded algebra which, as a KW –module, is isomorphic to the regular representation of W (see [Ch], p. 779), which we call the graded regular representation of the reflection datum 𝔾. We denote by R^{n}𝔾 the subspace of elements of degree n of R𝔾. Then we have (see [ShTo], p. 111) :

R𝔾 = ∑_{n=0}^{N^{∨}(𝔾)} R^{n}𝔾 .

The polynomial order.

The polynomial order of 𝔾 is the polynomial (in K[x]) denoted by |𝔾| and defined by the formula

(3.4)  |𝔾| := \frac{ε_{𝔾}x^{N(𝔾)}}{\frac{1}{|W |} ∑_{w∈W} \frac{1}{det_{V} (1 − xwϕ)^{∗}}} .

where det^{∗} denotes the complex conjugate of the determinant (Note that in this case, we have det_{V}^{∗} (1 − xwϕ) = det_{V} (1 − x(wϕ)^{−1})).

Comment. In the case of reductive groups, |𝔾|(q) is the order of G^{F}. See e.g. [BrMa1], th. 2.2.

3.5. Proposition. We have

|𝔾| = ε_{𝔾}x^{N(𝔾)} ∏_{j=1}^{j=r}(1 − ζ_{j}^{∗}x^{d_{j}}) = x^{N(𝔾)} ∏_{j=1}^{j=r}(x^{d_{j}} − ζ_{j}) ,

and in particular |𝔾| ∈ ℤ_{K}[x].

Proof of 3.5. By Molien’s formula (making free use of the notion of “graded character” of a graded module as in [BrMa1]) we have

\frac{1}{det_{V} (1 − xwϕ)} = tr(wϕ; SV) ,

Thus

\frac{1}{|W |} ∑_{w∈W} \frac{1}{det_{V} (1 − xwϕ)} = \frac{1}{|W |} ∑_{w∈W} tr(wϕ; SV)

= \frac{1}{|W |} ∑_{w∈W} tr(ϕw; SV) = tr(ϕ; (SV)^{W}) .

It results from the definition of generalized degrees that, as a ϕ-module, (SV)^{W} is isomorphic to K[P_{1}] ⊗ … ⊗ K[P_{r}] where the action of ϕ is given by ϕP_{j} = ζ_{j}P_{j}. We see that the first equality of the proposition results from this, and the second one results from the first one and from 1.2. □

Let us call “cyclotomic polynomials over K” the minimal polynomials over K of roots of unity. We see that an irreducible divisor of |𝔾| in K[x] is either x or a cyclotomic polynomial over K. More precisely,

3.6. Corollary.

(1) We have |𝔾| = x^{N(𝔾)} ∏_{Φ} Φ(x)^{a(Φ)} , where Φ runs over the set of cyclotomic polynomials over K, and where a(Φ) is the number of indices j such that ζ^{d_{j}} = ζ_{j} (where ζ is a chosen root of Φ).

(2) The degree of the polynomial |𝔾| is (N(𝔾) + N^{∨}(𝔾) + r).

(3) We have |𝔾|(1/x) = ε_{𝔾}x^{−(2N(𝔾)+N^{∨}(𝔾)+r)}|𝔾|(x)^{∗} .

Proof of 3.6. It is easy and left to the reader (use 1.2 again).

3D. Various constructions with reflection data.

Product of reflection data.

Let 𝔾 = (V, W ϕ) and 𝔾′ = (V′, W′ϕ′) be two reflection data. Their product is the reflection datum

𝔾 × 𝔾′ := (V × V′, (W × W′)(ϕ × ϕ′)) .

Comment. This corresponds to the product of reductive groups.

Extending scalars.

Let 𝔾 = (V, W ϕ) be a reflection datum on the subfield K of ℚ(μ_{∞}), and let K′ be an extension of K contained in ℚ(μ_{∞}). The reflection datum K′𝔾 is then defined as the image of 𝔾 under the functor K′ ⊗ · :

K′𝔾 := (K′ ⊗ V, K′ ⊗ (W ϕ)) .

Comment. This construction has no analogue for reductive groups.

Lifting scalars.

Let 𝔾 = (V, W ϕ) be a reflection datum and let a ∈ ℕ.

(3.7) We define the reflection datum 𝔾^{(a)} = (V^{(a)}, W^{(a)}ϕ^{(a)}) by the following rules: (ls.1) V^{(a)} := V × ⋯ × V (a times), and W^{(a)} := W × ⋯ × W (a times), (ls.2) ϕ^{(a)} is the product of ϕ (acting diagonally on V × ⋯ × V) by the a–cycle which

permutes cyclically the factors V of V^{(a)}.

We have

|𝔾^{(a)}|(x) = |𝔾|(x^{a}) .

Comment. This corresponds to an extension of scalars from F_{q} to F_{q^{a}} for reductive groups.

Changing x into ζx and generalized Ennola duality.

Let 𝔾 = (V, W ϕ) be a reflection datum. Let ζ ∈ K be a root of unity. We define the reflection datum 𝔾^{ζ} by

(3.8)  𝔾^{ζ} := (V, W ζϕ) .

From the definition of the polynomial order, we get

|𝔾^{ζ}|(x) = ε_{𝔾^{ζ}}ε_{𝔾}^{−1}ζ^{N(𝔾)}|𝔾|(ζ^{−1}x)

and, since

ε_{𝔾^{ζ}} = ε_{𝔾}ζ^{d_{1}+d_{2}+···+d_{r}} ,

we get (by 1.2)

|𝔾^{ζ}|(x) = ζ^{(N+N^{∨}+r)}|𝔾|(ζ^{−1}x) .

If ζId ∈ W, then 𝔾^{ζ} = 𝔾, and then all the invariants of 𝔾 and of 𝔾^{ζ} must coincide.

Comment. A prototype for this construction is the Ennola duality between the linear and unitary groups, which corresponds to changing x into −x in the reflection datum associated to linear groups.

The dual, the“semi–simple quotient” and adjoint reflection data.

• Let V^{∨} be the dual space of V. We still denote by W the image in GL(V^{∨}) of the group W acting through the contragredient representation w ↦ ^{t}w^{−1} and we set ϕ^{∨} := ^{t}ϕ^{−1}. Then the dual reflection datum of 𝔾 is

𝔾^{∨} := (V^{∨}, W ϕ^{∨}) .

Comment. This corresponds to the Langlands dual for algebraic groups. However, since Weyl groups are real reflection groups, the reflection data associated to an algebraic group and its Langlands dual are isomorphic.

• The “semi–simple quotient” of 𝔾 is “the quotient of 𝔾 by its center Z(𝔾)”, namely

𝔾_{ss} := (V/V^{W} , W ϕ) ,

where we identify W and ϕ with their images in the linear group of V/V^{W}.

Remark. One may also define the adjoint reflection datum of 𝔾 as

𝔾_{ad} := (((V^{∨})^{W})^{⊥}, W ϕ) ,

where we identify W and ϕ with their images in the linear group of ((V^{∨})^{W})^{⊥}. Denoting

by pr_{W} the projector of V defined by pr_{W} := \frac{1}{|W |} ∑_{w∈W} w , it is readily checked that

the application 1 − pr_{W} induces a W –isomorphism from V/V^{W} onto ((V^{∨})^{W})^{⊥}, which shows that 𝔾_{ss} and 𝔾_{ad} are isomorphic.

Comment. As the chosen names reflect, this corresponds to the quotient by the radical (resp. to the adjoint group) for reductive groups.

Intersection of a sub-reflection datum of maximal rank with a Levi sub-reflection datum.

Let 𝕃 = (V, W_{𝕃}vϕ) be a Levi sub-reflection datum of 𝔾 and let 𝕄 = (V, W_{𝕄}wϕ) be a sub-reflection datum of maximal rank of 𝔾.

(3.9) We say that 𝕃 ∩ 𝕄 is defined if

W_{𝕃}vϕ ∩ W_{𝕄}wϕ ≠ ∅ .

In that case, choosing an element u ∈ W_{𝕃}v ∩ W_{𝕄}w, we define

𝕃 ∩ 𝕄 := (V, (W_{𝕃} ∩ W_{𝕄})uϕ) .

Note that W_{𝕃}vϕ = W_{𝕃}uϕ and W_{𝕄}wϕ = W_{𝕄}uϕ, hence W_{𝕃}vϕ ∩ W_{𝕄}wϕ = (W_{𝕃} ∩ W_{𝕄})uϕ. Notice also that W_{𝕃} ∩ W_{𝕄} is a parabolic subgroup of W_{𝕄}. It is easy to see that 𝕃 ∩ 𝕄 is a well–defined (i.e., independent of the choice of u) Levi sub-reflection datum of 𝕄, and a sub-reflection datum of maximal rank of 𝕃.

By the preceding definition, it is clear that

(3.10) 𝕃 ∩ 𝕄 is defined if and only if there exists a maximal sub-torus 𝕋 of 𝔾 contained in both 𝕃 and 𝕄. In this case, 𝕋 is contained in 𝕃 ∩ 𝕄.

For w ∈ W and 𝕃 a sub–reflection datum of maximal rank of 𝔾, we denote by w𝕃 := w𝕃w^{−1} the conjugate of 𝕃 under w.

3.11. Definition. For 𝕃 a Levi sub-reflection datum and 𝕄 a sub-reflection datum of maximal rank of 𝔾, we denote by T_{W_{𝔾}}(𝕃, 𝕄) the set of all w ∈ W_{𝔾} such that ^{w}𝕃 ∩ 𝕄 is defined.

3E. Classification of reflection data.

Using the classification of finite irreducible complex reflection groups by Shephard and Todd [ShTo] it is possible to give a complete description of the possible reflection data as follows.

Let 𝔾 = (V, W ϕ) be a reflection datum. Let V_{1} ≤ V be a W ⟨ϕ⟩-invariant subspace of V and V_{2} a W ⟨ϕ⟩-invariant complement. Any reflection in W either acts trivially on V_{1} or on V_{2}. Thus, as W is generated by reflections, the decomposition V = V_{1} ⊕ V_{2} induces a ϕ-invariant direct product decomposition W = W_{1} × W_{2}, such that W_{i} acts trivially on V_{3−i} (i = 1, 2). This shows that 𝔾 is a direct product

𝔾 = 𝔾_{1} × 𝔾_{2}, with 𝔾_{i} = (V_{i}, W_{i}ϕ|_{V_{i}}) for i = 1, 2 .

Now assume that W ⟨ϕ⟩ acts irreducibly on V. Let V = V_{1}⊕…⊕V_{a} be the decomposition of V into W -irreducible subspaces. As above this induces a direct product decomposition W = W_{1} × … × W_{a} of W into reflection subgroups W_{i} such that W_{i} acts trivially on all V_{j} for j ≠ i. Moreover, the (V_{i}, W_{i}) are permuted transitively by ϕ. Thus we see that 𝔾 is the lifting of scalars

𝔾 = (V_{1}, W_{1}ψ)^{(a)}, with ψ = ϕ^{a}|_{V_{1}}

of the reflection datum (V_{1}, W_{1}ψ) where W_{1} acts irreducibly (hence absolutely irreducibly) on V_{1}.

It hence remains to classify those reflection data 𝔾 where V is an (absolutely) irreducible W -module. For this, it is useful to first determine normal embeddings of irreducible reflection groups in the same dimension.

3.12 Proposition. Let W be an irreducible finite complex reflection group on the complex vector space V and W′ a proper irreducible (normal) subgroup of W generated by a union of reflection classes of W, maximal in W with respect to these properties. Then (W, W′) are in the following list (where we adopt the notation of [ShTo] for irreducible reflection groups):

\begin{matrix}W & W′ \\ G(dep, e, n) & G(dep, ep, n), p prime \\ G_{5} & G_{4} \\ G_{6} & G(4, 2, 2), G_{4} \\ G_{7} & G_{6}, G_{5} \\ G_{8} & G(4, 2, 2) \\ G_{9} & G_{13}, G_{8} \\ G_{10} & G_{7}, G_{8} \\ G_{11} & G_{15}, G_{9}, G_{10} \\ G_{13} & G(4, 2, 2), G_{12} \end{matrix}  \begin{matrix}W & W′ \\ G_{14} & G_{12}, G_{5} \\ G_{15} & G_{13}, G_{14}, G_{7} \\ G_{17} & G_{22}, G_{16} \\ G_{18} & G_{20}, G_{16} \\ G_{19} & G_{21}, G_{17}, G_{18} \\ G_{21} & G_{22}, G_{20} \\ G_{26} & G(3, 3, 3), G_{25} \\ G_{28} & G(2, 2, 4) \end{matrix}

The proof is a routine case-by-case check, using for example the tables on p. 395 and p. 412 of [Co], and hence omitted.

3.13 Proposition. Let W be an irreducible complex reflection group on the n-dimen-sional complex vector space V and ϕ an automorphism of finite order of V normalizing W. Then up to multiples of the identity, either \overline{ϕ} = 1 or we are in one of the following cases :

- W = G(de, e, n), with e > 1, and ϕ̄ of order dividing e comes from the embedding G(de, e, n) < G(de, 1, n),

- W = G(4, 2, 2) and ϕ of order 3 comes from the embedding G(4, 2, 2) < G_{6},

- W = G(3, 3, 3) and ϕ of order 4 comes from the embedding G(3, 3, 3) < G_{26},

- W = G(2, 2, 4) and ϕ of order 3 comes from the embedding W (D_{4}) < W (F_{4}),

- W = G_{5} and ϕ of order 2 comes from the embedding G_{5} < G_{14},

- W = G_{7} and ϕ of order 2 comes from the embedding G_{7} < G_{10},

- W = G_{28} and ϕ realizes the graph-automorphism of W (F_{4}),

Here W (F_{4}), W (D_{4}), denote the Weyl groups of type F_{4}, D_{4} respectively.

Proof. First assume that W is imprimitive, so W = G(de, e, n) and clearly n ≥ 2. Then W has a unique system of imprimitivity unless

(de, e, n) ∈ {(2, 1, 2), (4, 2, 2), (3, 3, 3), (2, 2, 4)}

(see Lemma 2.7 in [Co], but note that there the case (4, 2, 2) was omitted since the number of orbits of W on P on p. 388 is gcd(e, 2)). Then as in the proof of [Co], Prop. 2.13, we may conclude that a finite subgroup of GU(V) containing W as a normal subgroup lies in ⟨G(e, 1, n), Z(GU(V))⟩ unless (de, e, n) is one of the above exceptions. It is easily seen that all exceptional automorphisms already arise from the embeddings given in Proposition 3.12 except for the case of G(2, 1, 2). But G(2, 1, 2) ≅ G(4, 4, 2) as reflection group, and the exceptional automorphism of the first group can be regarded as the automorphism of G(4, 4, 2) induced by the embedding into G(4, 1, 2).

For the primitive case first note that since the representation of W is absolutely irreducible, the center is left invariant by the normalizer of W in GL(V), so we can restrict ourselves to determining which non-trivial automorphisms of the projective groups W/Z(W) lift to W. If n = 2 then W/Z(W) is one of 𝔄_{4}, 𝔖_{4} or 𝔄_{5}. Only the first of these has outer automorphisms normalizing the degree 2 representation. Now W/Z(W) ≅ 𝔄_{4} precisely for G_{4}, …, G_{7}, but for G_{4} and G_{6} the extension of W/Z(W) does not lift to (V, W), so this leads to the given exceptions. In dimension n ≥ 3, the result is well known for the Coxeter groups (of types E_{i}, F_{4}, H_{3} and H_{4}), and for the remaining 9 cases is easily deduced from the identification of W with some well-known abstract finite group. (Note that both “exceptions” given in [Co, (5.14)] are in fact central extensions.) □

Note that all of these, except for ^{4}G(3, 3, 3), can be visualized by graph-automor-phisms of the corresponding diagram given in [BMR].

The polynomial orders of the twisted reflection data corresponding to the above automorphisms ϕ can easily be calculated from Proposition 3.5 in terms of the generalized degrees. For the infinite families ^{t}G(de, e, n), t|e, we have

|^{t}G(de, e, n)| = (x^{de} − 1)(x^{2de} − 1) … (x^{(n−1)de} − 1)(x^{nd} − ζ_{t}) ,

where ζ_{t} is a primitive t–th root of unity, while for the exceptional cases we obtain :

|^{3}G(4, 2, 2)| = (x^{4} − 1)(x^{4} − ζ_{3}^{2})
|^{4}G(3, 3, 3)| = (x^{6} − 1)(x^{6} + 1)
|^{3}G(2, 2, 4)| = (x^{2} − 1)(x^{6} − 1)(x^{8} + x^{4} + 1)
|^{2}G_{5}| = (x^{6} − 1)(x^{12} + 1)
|^{2}G_{7}| = (x^{12} − 1)(x^{12} + 1)
|^{2}G_{28}| = (x^{2} − 1)(x^{6} + 1)(x^{8} − 1)(x^{12} + 1)

(notice that ^{3}G(2, 2, 4) = ^{3}D_{4} and ^{2}G_{28} = ^{2}F_{4}).

4. Uniform class functions on a reflection datum

4A. Generalities, induction and restriction.

From now on, we assume that K splits the reflection datum 𝔾 = (V, W ϕ), i.e., that K contains all roots of unity with order the l.c.m. of the orders of the elements of W ϕ.

Let CF_{uf}(𝔾) be the ℤ_{K}–module of all W –invariant functions on the coset W ϕ with values in ℤ_{K}, called uniform class functions on 𝔾. For α ∈ CF_{uf}(𝔾), we denote by α^{∗} its complex conjugate.

If α and α′ ∈ CF_{uf}(𝔾), we set ⟨α, α′⟩_{𝔾} := \frac{1}{|W |} ∑_{w∈W} α(wϕ)α′(wϕ)^{∗} .

Notation.

• If ℤ_{K} → O is a ring morphism, we denote by CF_{uf}(𝔾, O) the O–module of W – invariant functions on W ϕ with values in O, which we call the module of uniform class functions on 𝔾 with values in O. We have CF_{uf}(𝔾, O) = O ⊗_{ℤ_{K}} CF_{uf}(𝔾).

• For wϕ ∈ W ϕ, we denote by ch_{wϕ}^{𝔾} (or simply ch_{wϕ}) the characteristic function of

the orbit of wϕ under W. The family (ch_{wϕ}^{𝔾}) (where wϕ runs over a complete set of representatives of the orbits of W on W ϕ) is a basis of CF_{uf}(𝔾).

• For wϕ ∈ W ϕ, we set

R_{wϕ}^{𝔾} := |C_{W} (wϕ)|ch_{wϕ}^{𝔾}

(or simply R_{wϕ}). The ℤ_{K}–module generated by the functions R_{wϕ}^{𝔾} is denoted by CF_{uf}^{pr}(𝔾). For α ∈ CF_{uf}(𝔾) and wϕ ∈ W ϕ we have ⟨α, R_{wϕ}⟩_{𝔾} = α(wϕ), so CF_{uf}(𝔾) is the ℤ_{K}-dual of CF_{uf}^{pr}(𝔾).

[The exponent “pr” stands for “projective”, by analogy with the vocabulary of modular representation theory of finite groups]

Comment. In the case of reductive groups, we have K = ℚ. Let Uch(G^{F}) be the set of unipotent characters of G^{F}: then the map which associates to R𝔾_{wϕ} the Deligne–Lusztig character R_{T_{wF}}^{G} (Id) defines an isometric embedding (for the scalar products ⟨α, α′⟩_{𝔾} and ⟨α, α′⟩_{G^{F}}) from CF_{uf}(𝔾) onto the sub-ℤ-module of ℚ Uch(G^{F}) of the ℚ-linear combinations of Deligne-Lusztig characters (i.e. “unipotent uniform functions”) which have an integral scalar product with the Deligne-Lusztig characters.

• Let ⟨W ϕ⟩ be the subgroup of GL(V) generated by W ϕ. We recall that we denote by \overline{ϕ} the image of ϕ in ⟨W ϕ⟩/W — thus the group ⟨W ϕ⟩/W is cyclic and generated by \overline{ϕ}.

For ψ ∈ Irr(⟨W ϕ⟩), we denote by R_{ψ}^{𝔾} (or simply R_{ψ}) the restriction of ψ to the coset W ϕ. We have

R_{ψ}^{𝔾} = \frac{1}{|W |} ∑_{w∈W} ψ(wϕ)R_{wϕ}^{𝔾} ,

and we call such a function a uniform almost character of 𝔾.

Let Irr(W)\overline{ϕ} denote the set of \overline{ϕ}–stable irreducible characters of W.

For χ ∈ Irr(W)\overline{ϕ}, we denote by E_{𝔾}(χ) (or simply E(χ)) the set of restrictions to W ϕ of the extensions of χ to characters of ⟨W ϕ⟩. If K contains the δ(𝔾)–th roots of unity (where δ(𝔾) denotes the order of the twist \overline{ϕ}), the group ⟨\overline{ϕ}⟩ acts regularly on E(χ).

Each element of E_{𝔾}(χ) has norm 1, the sets E_{𝔾}(χ) for χ ∈ Irr(W)\overline{ϕ} are mutually orthogonal, and we have

CF_{uf}(𝔾, K) = ⨁_{χ∈Irr(W)\overline{ϕ}}^{⊥} KE_{𝔾}(χ) ,

where we set KE_{𝔾}(χ) := KR_{ψ}^{𝔾} for some (any) ψ ∈ E_{𝔾}(χ).

Induction and restriction.

Let 𝕃 = (V, W_{𝕃}wϕ) be a sub-reflection datum of maximal rank of 𝔾, and let α ∈ CF_{uf}(𝔾) and β ∈ CF_{uf}(𝕃).

We denote • by Res_{𝕃}^{𝔾} α the restriction of α to the coset W_{𝕃}wϕ,

• by Ind_{𝕃}^{𝔾}β the uniform class function on 𝔾 defined by

(4.1)  Ind_{𝕃}^{𝔾}β(uϕ) := \frac{1}{|W_{𝕃}|} ∑_{v∈W} β̃(vuϕv^{−1}) for uϕ ∈ W_{𝔾}ϕ,

where β̃(xϕ) = β(xϕ) if x ∈ W_{𝕃}w, and β̃(xϕ) = 0 if x ∉ W_{𝕃}w . In other words, we have

(4.2)  Ind_{𝕃}^{𝔾}β(uϕ) = ∑_{v∈W_{𝔾}/W_{𝕃},^{v}(uϕ)∈W_{𝕃}wϕ} β(^{v}(uϕ)) .

For any reflection datum 𝔾 we denote by 1^{𝔾} the constant function on W ϕ with value 1. For w ∈ W, let us denote by 𝕋_{wϕ} the maximal torus of 𝔾 defined by 𝕋_{wϕ} := (V, wϕ) .

It follows from the definitions that

(4.3)  R_{wϕ}^{𝔾} = Ind_{𝕋_{wϕ}}^{𝔾}1^{𝕋_{wϕ}} .

We have the Frobenius reciprocity:

(4.4)  ⟨α, Ind_{𝕃}^{𝔾}β⟩_{𝔾} = ⟨Res_{𝕃}^{𝔾} α, β⟩_{𝕃} .

Comment. In the case of reductive groups, assume that 𝕃 is a Levi sub-reflection datum. Then Ind_{𝕃}^{𝔾} corresponds to Lusztig induction from L to G (this results from definition 4.1 applied to a Deligne–Lusztig character which, using the transitivity of Lusztig induction, agrees with Lusztig induction). Similarly, the Lusztig restriction of a uniform function is uniform by [DeLu], theorem 7, so by (4.4) Res_{𝕃}^{𝔾} corresponds to Lusztig restriction.

When 𝕃 corresponds to a reductive subgroup of maximal rank which is not a Levi subgroup, then Ind_{𝕃}^{𝔾} corresponds to a generalization of Lusztig induction to that setting. However, this generalization does not necessarily map ℤ Uch(L^{F}) to ℤ Uch(G^{F}). For instance, when L is of type A_{2} corresponding to the long roots of G of type G_{2}, then the image of ℤ Uch(L^{F}) by this generalized induction is only in ℤ[1/3] Uch(G^{F}).

The Mackey formula.

Let now 𝕃 be a Levi sub-reflection datum and let 𝕄 be a sub-reflection datum of maximal rank of 𝔾.

It is clear that W_{𝕃} acts on T_{W_{𝔾}}(𝕃, 𝕄) (see definition 3.11 above) from the right, while W_{𝕄} acts on T_{W_{𝔾}}(𝕃, 𝕄) from the left. If we let w run over a chosen double coset W_{𝕄}vW_{𝕃} for some v ∈ T_{W_{𝔾}}(𝕃, 𝕄), we see that ^{w}𝕃 ∩ 𝕄 is defined up to W_{^{w}𝕃}–conjugation as a sub-reflection datum of ^{w}𝕃 and up to W_{𝕄}–conjugation as a sub-reflection datum of 𝕄, which proves that the operations Ind_{𝕄∩^{w}𝕃}^{𝕄} and Res_{𝕄∩^{w}𝕃}^{^{w}𝕃} depend only on the double coset of w. This gives sense to the following formula

(4.5)  Res_{𝕄}^{𝔾} ·Ind_{𝕃}^{𝔾} = ∑_{w∈W_{𝕄}\T_{W_{𝔾}} (𝕃,𝕄)/W_{𝕃}} Ind_{𝕄∩^{w}𝕃}^{𝕄} · Res_{𝕄∩^{w}𝕃}^{^{w}𝕃} ·ad(w) ,

whose proof goes like the proof of the Mackey formula for ordinary induction and restriction (note that T_{W_{𝔾}}(𝕃, 𝕄) may be empty).

Comment. In the case of reductive groups, assuming that both 𝕃 and 𝕄 are Levi subreflection data, the Mackey formula corresponds to the Mackey formula for Lusztig induction and restriction (projected on uniform function).

Uniform class functions on 𝔾^{ζ}.

The map σ_{𝔾}^{ζ}: CF_{uf}(𝔾) → CF_{uf}(𝔾^{ζ}), given by σ_{𝔾}^{ζ}α(wζϕ) = α(wϕ), is an isometry. The map 𝕃 ↦ 𝕃^{ζ} is a W_{𝔾}–equivariant bijection from the set of all sub-reflection data of maximal rank of 𝔾 (resp. of all Levi sub–reflection data of 𝔾) onto the set of all sub-reflection data of maximal rank of 𝔾^{ζ} (resp. of all Levi sub–reflection data of 𝔾^{ζ}). It is clear that

(4.6)  σ_{𝔾}^{ζ} · Ind_{𝕃}^{𝔾} = Ind_{𝕃^{ζ}}^{𝔾^{ζ}} · σ_{𝕃}^{ζ} , σ_{𝕃}^{ζ} · Res_{𝕃}^{𝔾} = Res_{𝕃^{ζ}}^{𝔾^{ζ}} ·σ_{𝔾}^{ζ} .

Uniform class functions on 𝔾^{(a)}.

Let a ∈ ℕ. We have (see 3.7) W_{𝔾^{(a)}} = (W_{𝔾})^{a}, and it is clear that the map (w_{1}, w_{2}, …, w_{a})ϕ^{(a)} ↦ w_{1}w_{2} ⋯ w_{a}ϕ defines a bijection between the set of classes of W_{𝔾^{(a)}}ϕ^{(a)} under W_{𝔾^{(a)}}–conjugacy and the set of classes of W_{𝔾}ϕ under W_{𝔾}–conjugacy. Thus it induces an isometry

σ_{𝔾}^{(a)} : CF_{uf}(𝔾^{(a)}) ⟶^{∼} CF_{uf}(𝔾) .

4.7. Proposition. We have

σ_{𝔾}^{(a)} · Ind_{𝕃^{(a)}}^{𝔾^{(a)}} = Ind_{𝕃}^{𝔾} · σ_{𝕃}^{(a)}
Res_{𝕃}^{𝔾} ·σ_{𝔾}^{(a)} = σ_{𝕃}^{(a)} · Res_{𝕃^{(a)}}^{𝔾^{(a)}} .

4B. Degrees.

• We denote by tr_{R𝔾} the uniform class function on 𝔾 (with values in the polynomial ring ℤ_{K}[x]) defined by the character of the “graded regular representation” R𝔾 (see above). Thus the value of the function tr_{R𝔾} on wϕ is

tr_{R𝔾}(wϕ) := ∑_{n=0}^{N^{∨}(𝔾)} tr(wϕ; R^{n}𝔾)x^{n} .

We call tr_{R𝔾} the regular character of 𝔾.

• We define the degree, a linear function

Deg_{𝔾}: CF_{uf}(𝔾) → K[x] ,

as follows : for α ∈ CF_{uf}(𝔾), we set

(4.8)  Deg_{𝔾}(α) := ⟨α, tr_{R𝔾}⟩_{𝔾} = ∑_{n=0}^{N^{∨}(𝔾)} (\frac{1}{|W |} ∑_{w∈W} α(wϕ)tr(wϕ; R^{n}𝔾)^{∗}) x^{n} .

We shall often omit the subscript 𝔾 (writing then Deg(α)) when the context allows it. Notice that

(4.9)  Deg(R_{wϕ}^{𝔾}) = tr_{R𝔾}(wϕ) ,

and so in particular that

(4.10)  Deg(R_{wϕ}^{𝔾}) ∈ ℤ_{K}[x] .

Degrees of almost characters.

Let E be a K⟨W ϕ⟩–module. Let χ_{E} be the restriction of the character of E (a uniform class function on ⟨W ϕ⟩) to W ϕ. Then the degree of R_{χ_{E}} is the “graded multiplicity” of E in the graded regular representation R𝔾 :

(4.11)  Deg_{𝔾}(R_{χ_{E}}) = tr(ϕ; Hom_{KW} (R𝔾, E)) .

Notice that

(4.12)  Deg_{𝔾}(R_{χ_{E}}) ∈ ℤ[exp^{2iπ/δ(𝔾)}][x] ,

(we recall that δ(𝔾) is the order of the twist \overline{ϕ} of 𝔾).

Remark. Suppose 𝔾 is split. Let χ ∈ Irr(W). Then χ is a uniform function on 𝔾, and we have (see above §1, 1.6)

Deg_{𝔾}(R_{χ}) = Feg_{χ}(x) .

Let χ ∈ Irr(W)\overline{ϕ}. If ψ ∈ Irr(⟨W ϕ⟩) runs over the set of extensions of χ to ⟨W ϕ⟩, and if K contains the δ(𝔾)–th roots of unity, the set of degrees Deg(R_{ψ}) is an orbit of μ_{δ(𝔾)}(K) on ℤ_{K}[x]. The element Deg_{𝔾}(R_{ψ})^{∗} · R_{ψ} depends only on χ and is the orthogonal projection of tr_{R𝔾} onto K[x]E_{𝔾}(χ). We set

(4.13)  Reg_{χ}^{𝔾} := Deg_{𝔾}(R_{ψ})^{∗} · R_{ψ} ,

and in other words, we have

(4.14)  tr_{R𝔾} = ∑_{χ∈Irr(W)\overline{ϕ}} Reg_{χ}^{𝔾} .

4.15. Lemma. We have

tr_{R𝔾} = \frac{1}{|W |} ∑_{w∈W} Deg_{𝔾}(R_{wϕ}^{𝔾})^{∗}R_{wϕ}^{𝔾} ,

and in particular

Deg_{𝔾}(tr_{R𝔾}) = \frac{|𝔾||𝔾|^{∗}}{x^{2N(𝔾)}} \frac{1}{|W |} ∑_{w∈W} \frac{1}{det_{V} (1 − xwϕ)det_{V} (1 − xwϕ)^{∗}} .

Proof of 4.15. The first formula is an immediate consequence of the definition of R_{wϕ}^{𝔾} and of 4.9, while the second one is a consequence of the first one. □

Polynomial order and degrees.

We have (cf. for example [Bou], chap. V, §5, th. 2) the following isomorphism of ⟨W ϕ⟩–modules :

SV ≃ R𝔾 ⊗_{K} (SV)^{W} ,

from which we deduce

(4.16)  \frac{1}{|W |} ∑_{w∈W} \frac{α(wϕ)}{det_{V} (1 − xwϕ)^{∗}} = Deg(α)\frac{1}{|W |} ∑_{w∈W} \frac{1}{det_{V} (1 − xwϕ)^{∗}} ,

or, in other words

(4.17)  ⟨α, tr_{SV} ⟩_{𝔾} = ⟨α, tr_{R𝔾}⟩_{𝔾}⟨1^{𝔾}, tr_{SV} ⟩_{𝔾} ,

where 1^{𝔾} is the constant function 1 on W_{𝔾}ϕ.

Let us set S_{𝔾}(α) := ⟨α, tr_{SV} ⟩_{𝔾} . Then 4.17 becomes :

(4.18)  S_{𝔾}(α) = Deg_{𝔾}(α)S_{𝔾}(1^{𝔾}) .

By 3.4 we get

|𝔾| = ε_{𝔾} \frac{x^{N(𝔾)}}{S_{𝔾}(1^{𝔾})} , hence
(4.19)  S_{𝔾}(α)|𝔾| = ε_{𝔾}x^{N(𝔾)}Deg_{𝔾}(α) .

• For a sub-reflection datum 𝕃 of maximal rank of 𝔾, by the Frobenius reciprocity (4.4), we have

(4.20)  Deg(Ind_{𝕃}^{𝔾}1^{𝕃}) = ⟨1^{𝕃}, Res_{𝕃}^{𝔾} tr_{R𝔾}⟩_{𝕃} = ∑_{n=0}^{N^{∨}(𝔾)} tr(wϕ; (R^{n}𝔾)^{W_{𝕃}})^{∗}x^{n} .

where W_{𝕃}wϕ is the coset associated to 𝕃 and (R^{n}𝔾)^{W_{𝕃}} are the W_{𝕃}–invariants in R^{n}𝔾.

4.21. Proposition. We have

|𝔾|/|𝕃| = ε_{𝔾}ε_{𝕃}^{−1}x^{N(𝔾)−N(𝕃)}Deg(Ind_{𝕃}^{𝔾}1^{𝕃}) ,

and in particular |𝕃| divides |𝔾| (in ℤ_{K}[x]).

Proof of 4.21. By 4.19 the proposition is equivalent to

S_{𝔾}(Ind_{𝕃}^{𝔾}1^{𝕃}) = S_{𝕃}(1^{𝕃})

which results from Frobenius reciprocity (4.4). □

• Let us recall that every element wϕ ∈ W ϕ defines a maximal torus (or, equivalently, a minimal Levi sub-reflection datum) 𝕋_{wϕ} := (V, wϕ) . By 4.3 and 4.9, we have

tr_{R𝔾}(wϕ) = Deg(Ind_{𝕋_{wϕ}}^{𝔾}1^{𝕋_{wϕ}})^{∗} .

So

(4.22)  |𝔾|/|𝕋_{wϕ}| = ε_{𝔾}ε_{𝕋_{wϕ}}^{−1}x^{N(𝔾)}tr_{R𝔾}(wϕ)^{∗} .

It follows from 4.21 and 4.22 that

(4.23)  Res_{𝕃}^{𝔾} tr_{R𝔾}^{∗} = Deg_{𝔾}(Ind_{𝕃}^{𝔾}1^{𝕃})tr_{R𝕃}^{∗} .

(4.24) For β ∈ CF_{uf}(𝕃), we have Deg_{𝔾}(Ind_{𝕃}^{𝔾}β) = Deg_{𝔾}(Ind_{𝕃}^{𝔾}1^{𝕃})Deg_{𝕃}(β) .

Indeed, Deg_{𝔾}(Ind_{𝕃}^{𝔾}β) = ⟨β, Res_{𝕃}^{𝔾} tr_{R𝔾}⟩_{𝕃} = Deg_{𝔾}(Ind_{𝕃}^{𝔾}1^{𝕃})⟨β, tr_{R𝕃}⟩_{𝕃} .

Comment. In the case of reductive groups, it follows from 4.9 and 4.22 that Deg(R_{wϕ})(q) is the degree of the Deligne–Lusztig character R_{T_{wϕ}}^{G}. Since the regular representation of G(q) is uniform, it follows that tr_{RG} corresponds to a (graded by x) version of the unipotent part of the regular representation of G^{F}, and that Deg corresponds indeed to the (generic) degree for unipotent uniform functions on G(q).

Changing x to 1/x.

As a particular uniform class function on 𝔾, we can consider the function det_{V} restricted to W_{𝔾}ϕ, which we still denote by det_{V}. Notice that this restriction might also be denoted by R_{det_{V}}^{𝔾}, since it is the almost character associated to the character of ⟨W ϕ⟩ defined by det_{V}.

4.25. Proposition. Let α be a uniform class function on 𝔾. We have

S_{𝔾}(αdet_{V}^{∗})(x) = (−1)^{r}x^{−r}S_{𝔾}(α^{∗})(1/x)^{∗}
Deg_{𝔾}(αdet_{V}^{∗}) = (−1)^{r}ε_{𝔾}^{∗}x^{N^{∨}(𝔾)}Deg_{𝔾}(α^{∗})(1/x)^{∗} .

Proof of 4.25. On one hand, it is readily checked that

S_{𝔾}(αdet_{V}^{∗})(x) = (−1)^{r}x^{−r}S_{𝔾}(α^{∗})(1/x)^{∗} .

On the other hand by 4.19, we know that S_{𝔾}(α)|𝔾| = ε_{𝔾}x^{N(𝔾)}Deg(α), and by 3.6, 3, we know that |𝔾|(x) = ε_{𝔾}x^{2N(𝔾)+N^{∨}(𝔾)+r}|𝔾|(1/x)^{∗} . This is enough to prove 4.25. □

4.26. Corollary. We have

Deg_{𝔾}(det_{V}^{∗}) = (−1)^{r}ε_{𝔾}^{∗}x^{N^{∨}(𝔾)}
Deg_{𝔾}(det_{V}) = (−1)^{r}ε_{𝔾}x^{N(𝔾)} .

Proof of 4.26.

The first equality is an immediate consequence of 4.25, since Deg(1) = 1.

It follows from 4.11 that the second equality is equivalent to the following two assertions :

(a) the representation det_{V} occurs in degree N(𝔾) in the graded regular representation R𝔾,

(b) the trace of ϕ on the representation det_{V} of W is equal to (−1)^{r}ε_{𝔾}.

Let us prove (a). For H ∈ A a reflecting hyperplane, there is only one line L_{H} which is stable by the parabolic subgroup W_{H}, and such that H ⊕ L_{H} = V. For each H ∈ A we choose a non trivial element v_{H} in L_{H}. Then the element δ := ∏_{H∈A} v_{H} has degree N, and it follows from [Co], 1.8, that w.δ = det_{V} (w)δ for all w ∈ W, which proves (a).

Let us prove (b). Since Deg(det_{V}^{∗}) = (−1)^{r}ε_{𝔾}^{∗}x^{N^{∨}(𝔾)}, we see that the trace of ϕ on the representation det_{V}^{∗} of W is equal to (−1)^{r}ε_{𝔾}^{∗}. By duality we get (b). □

5. Φ–Sylow theory and Φ–split Levi sub-reflection data

5A. The Sylow theorems.

Let 𝕋 = (V, wϕ) be a maximal torus. It is immediate to check that in this case the polynomial order |𝕋| is det_{V} (x − wϕ) .

Let Φ(x) ∈ K[x] be a cyclotomic polynomial. A Φ–reflection datum is a torus whose polynomial order is a power of Φ.

5.1. Theorem. Let 𝔾 be a reflection datum over K and let Φ be a cyclotomic polynomial on K.

(1) If Φ divides |𝔾|, there exist non trivial Φ–sub-reflection data of 𝔾.

(2) Let 𝕊 be a maximal Φ–sub-reflection datum of 𝔾. Then |𝕊| = Φ^{a(Φ)}, contribution of Φ to |𝔾|.

(3) Two maximal Φ–sub-reflection data of 𝔾 are conjugate under W_{𝔾}.

(4) Let 𝕊 be a maximal Φ–sub-reflection datum of 𝔾. We set 𝕃 := C_{𝔾}(𝕊) and W_{𝔾}(𝕃) := N_{W_{𝔾}}(𝕃)/W_{𝕃} . Then

|𝔾|/(|W_{𝔾}(𝕃)||𝕃|) ≡ 1 mod Φ .

The maximal Φ–sub-reflection data of 𝔾 are called the Sylow Φ–sub-reflection data. Before proving 5.1, let us state and comment the following important supplementary

property.

5.2. Theorem. Let 𝕊 be a maximal Φ–sub-reflection datum of 𝔾 and let 𝕃 := C_{𝔾}(𝕊). If 𝕃 = (V, W_{𝕃}wϕ), we set V (𝕃, Φ) := ker Φ(wϕ) ∩ V^{W_{𝕃}}, viewed as a vector space over the field K[x]/Φ(x) through its natural structure of K[wϕ]–module. Then the pair (V (𝕃, Φ), W_{𝔾}(𝕃)) is a reflection group.

Comment 5.2. The assertion (5) is a slight reformulation of the main result of [LeSp]. It had first been proved by Springer [Sp] in the particular case where 𝕃 is a torus (i.e., with the notation of §5.B, when wϕ is a regular element), and it had been checked case by case for many groups (see [BMM1]).

It should be noticed that, at least for W spetsial (see [BMM2]), 5.2 is a particular case of a more general result involving normalizers of “cuspidal Φ–pairs”.

Proof of 5.1.

As we shall see, assertions (1) to (3) are consequences of the main results of Springer in [Sp] (see also theorem 3.4 in [BrMa1]).

For each cyclotomic polynomial Φ over K and w ∈ W, we denote by V (wϕ, Φ) the kernel of the endomorphism Φ(wϕ). So (V (wϕ, Φ), (wϕ)_{|_{V (wϕ,Φ)}}) is a torus of 𝔾, denoted here by 𝕊(wϕ, Φ).

Let us denote by K′ a Galois extension of K which splits 𝔾. For every root ζ of Φ in K′, we set V′(wϕ, ζ) := V′(wϕ, (x − ζ)). It is clear that V′(wϕ, Φ) = ⨁_{ζ} V′(wϕ, ζ) where ζ runs over the set of roots of Φ, and that dimV′(wϕ, Φ) = deg(Φ).dimV′(wϕ, ζ) . It follows from [Sp], 3.4 and 6.2, that

(S1) max_{(w∈W)} dimV′(wϕ, ζ) = a(Φ),

(S2) for all w ∈ W, there exists w′ ∈ W such that dimV′(w′ϕ, ζ) = a(Φ) and

V′(wϕ, ζ) ⊂ V′(w′ϕ, ζ), (S3) if w_{1}, w_{2} ∈ W are such that dimV′(w_{1}ϕ, ζ) = dimV′(w_{2}ϕ, ζ) = a(Φ), there exists

w ∈ W such that w · V′(w_{1}ϕ, ζ) = V′(w_{2}ϕ, ζ).

(S1) shows that if a(Φ) ≠ 0, there exists w ∈ W such that the rank of 𝕊(wϕ, Φ) is a(Φ) deg(Φ), which implies the first assertion of theorem 5.1.

If V′(wϕ, ζ) ⊆ V′(w′ϕ, ζ) and σ ∈ Gal(K′/K) we have V′(wϕ, σ(ζ)) ⊆ V′(w′ϕ, σ(ζ)), hence

• V′(wϕ, Φ) ⊂ V′(w′ϕ, Φ),

• w′ϕ_{|_{V (wϕ,Φ)}} = wϕ_{|_{V (wϕ,Φ)}}.

So (S2) implies that for all w ∈ W, there exists w′ ∈ W such that the rank of 𝕊(w′ϕ, Φ) is a(Φ) deg(Φ) and 𝕊(wϕ, Φ) is contained in 𝕊(w′ϕ, Φ), which proves assertion (2) of theorem 5.1.

For the same reason, (S3) shows that if w_{1} and w_{2} are two elements of W such that both 𝕊(w_{1}ϕ, Φ) and 𝕊(w_{2}ϕ, Φ) have rank a(Φ) deg(Φ), there exists w ∈ W such that 𝕊(w^{−1}w_{1}ϕw, Φ) = 𝕊(w_{2}ϕ, Φ), which proves assertion (3) of theorem 5.1.

The proof of the fourth assertion requires several steps.

5.3. Lemma. Let 𝕊 be a maximal Φ–sub-reflection datum of 𝔾, and set 𝕃 = C_{𝔾}(𝕊). Let α be a class function on 𝔾. We have :

|𝔾|S_{𝔾}(α) ≡ \frac{|𝔾|}{|𝕃||W_{𝔾}(𝕃)|}|𝕃|S_{𝕃}(Res_{𝕃}^{𝔾}(α)) mod Φ ,

ε_{𝔾}x^{N(𝔾)}Deg_{𝔾}(α) ≡ \frac{|𝔾|}{|W_{𝔾}(𝕃)||𝕃|}ε_{𝕃}x^{N(𝕃)}Deg_{𝕃}(Res_{𝕃}^{𝔾} α) mod Φ .

Proof of 5.3. By 4.19, we have

|𝔾|S_{𝔾}(α) = \frac{1}{|W |} ∑_{w∈W} α(wϕ)\frac{|𝔾|}{det(1 − xwϕ)^{∗}} .

Since

\frac{|𝔾|}{det(1 − xwϕ)^{∗}} = (−1)^{r}det(wϕ)\frac{|𝔾|}{|𝕋_{wϕ}|} ,

it follows from the first three assertions of theorem 5.1 that

|𝔾|S_{𝔾}(α) ≡ \frac{|𝔾|}{|W |} ∑_{w} \frac{α(wϕ)}{det(1 − xwϕ)^{∗}} mod Φ ,

where w runs over those elements of W such that V (wϕ, Φ) is of maximal dimension. These subspaces are permuted transitively by W (cf. (S3) above). Let V (w_{0}ϕ, Φ) be one of them, and let 𝕊 be the Φ–sub-reflection datum defined by

𝕊 = (V (w_{0}ϕ, Φ), (w_{0}ϕ)_{|_{V (w_{0}ϕ,Φ)}}) .

We set 𝕃 := C_{𝔾}(𝕊) =: (V, W_{𝕃}w_{0}ϕ). The group N_{W} (𝕊) is the set of all w ∈ W such that w.V (w_{0}ϕ, Φ) = V (w_{0}ϕ, Φ) and (ww_{0}ϕw^{−1})_{|_{V (w_{0}ϕ,Φ)}} = (w_{0}ϕ)_{|_{V (w_{0}ϕ,Φ)}} . Since in this case every w ∈ W such that w.V (w_{0}ϕ, Φ) = V (w_{0}ϕ, Φ) belongs to N_{W} (𝕊), we have

|𝔾|S_{𝔾}(α) ≡ |W : N_{W} (𝕊)|\frac{|𝔾|}{|W |} ∑_{w∼w_{0}} \frac{α(wϕ)}{det(1 − xwϕ)^{∗}} mod Φ ,

where “w ∼ w_{0}” means that V (wϕ, Φ) = V (w_{0}ϕ, Φ).

• Following (S3) above, the elements w ∼ w_{0} are those such that w_{0}^{−1}w acts trivially on V (wϕ_{0}, Φ), i.e., are the elements of the coset w_{0}W_{𝕃}. Thus

∑_{w∼w_{0}} \frac{α(wϕ)}{det(1 − xwϕ)^{∗}} = ∑_{w∈W_{𝕃}} \frac{α(ww_{0}ϕ)}{det(1 − xww_{0}ϕ)^{∗}} = |W_{𝕃}|S_{𝕃}(Res_{𝕃}^{𝔾}(α)) .

• We know that N_{W} (𝕊) ⊂ N_{W} (𝕃). Let us check the inverse inclusion. The group N_{W} (𝕃)/W_{𝕃} acts on V^{W_{𝕃}} and centralizes (w_{0}ϕ)_{|_{V ^{W_{𝕃}}}}. Hence it stabilizes the characteristic subspaces of (w_{0}ϕ)_{|_{V ^{W_{𝕃}}}}, among them V (w_{0}ϕ, Φ). This shows that N_{W} (𝕃) = N_{W} (𝕊). In particular |N_{W} (𝕊)| = |W_{𝕃}||W_{𝔾}(𝕃)|.

The first congruence of lemma 5.3 results from what precedes.

The second congruence results from the equality (see 4.19 above)

|𝔾|S_{𝔾}(α) = ε_{𝔾}x^{N(𝔾)}Deg_{𝔾}(α) .

□

Applying the preceding lemma 5.3 to the cases where α is 1^{𝔾} or det_{V}, and applying 4.26, we get

ε_{𝔾}x^{N(𝔾)} ≡ \frac{|𝔾|}{|W_{𝔾}(𝕃)||𝕃|}ε_{𝕃}x^{N(𝕃)} mod Φ ,

(ε_{𝔾}x^{N(𝔾)})^{2} ≡ \frac{|𝔾|}{|W_{𝔾}(𝕃)||𝕃|} (ε_{𝕃}x^{N(𝕃)})^{2} mod Φ .

The congruence

\frac{|𝔾|}{|W_{𝔾}(𝕃)||𝕃|} ≡ 1 mod Φ

is an easy consequence of the previous two congruences. □

We may notice that what precedes also proves that

ε_{𝔾}x^{N(𝔾)} ≡ ε_{𝕃}x^{N(𝕃)} mod Φ ,

which, applied to 5.3, gives

5.4. Proposition. Let 𝕊 be a maximal Φ–sub-reflection datum of 𝔾, and set 𝕃 = C_{𝔾}(𝕊). Let α be a class function on 𝔾. We have :

Deg_{𝔾}(α) ≡ Deg_{𝕃}(Res_{𝕃}^{𝔾}(α)) mod Φ .

In particular, we have

ε_{𝔾}x^{N(𝔾)} ≡ ε_{𝕃}x^{N(𝕃)} mod Φ ,
ε_{𝔾}^{∗}x^{N^{∨}(𝔾)} ≡ ε_{𝕃}^{∗}x^{N^{∨}(𝕃)} mod Φ ,
x^{N(𝔾)+N^{∨}(𝔾)} ≡ x^{N(𝕃)+N^{∨}(𝕃)} mod Φ .

Φ–split Levi sub-reflection data.

We call Φ–split Levi sub-reflection data the centralizers in 𝔾 of the Φ–sub-reflection data of 𝔾. In particular, the centralizers of the Sylow Φ–sub-reflection data are the minimal Φ–split Levi sub-reflection data. It then follows that the preceding congruence holds for every Φ–split Levi sub-reflection datum of 𝔾, from which we deduce by 4.21 :

(5.5) For any Φ–split Levi sub-reflection datum 𝕃 of 𝔾, we have

|𝔾|/|𝕃| ≡ Deg(Ind_{𝕃}^{𝔾}1^{𝕃}) mod Φ .

Given a Levi sub-reflection datum (V, W_{𝕃}wϕ), we define its image in 𝔾_{ss} (resp. its image in 𝔾_{ad}) to be (V/V^{W} , W_{𝕃}wϕ) (resp. (((V^{∨})^{W})^{⊥}, W_{𝕃}wϕ)).

5.6. Lemma. A Levi sub-reflection datum is Φ-split if and only if its image in 𝔾_{ss} (resp. in 𝔾_{ad}) is Φ–split.

Mackey formula for Φ–split sub-reflection data.

The Mackey formula becomes particularly simple when we restrict ourselves to pairs of Φ–split Levi sub-reflection data.

5.7. Proposition. Let 𝕄_{1} and 𝕄_{2} be two Φ–split Levi sub-reflection data of 𝔾.

(1) 𝕄_{1} ∩ 𝕄_{2} is defined if and only if 𝕄_{1} and 𝕄_{2} contain a common Sylow Φ–subreflection datum of 𝔾. In particular T_{W_{𝔾}}(𝕄_{1}, 𝕄_{2}) ≠ ∅.

(2) If 𝕄_{1} ∩ 𝕄_{2} is defined, then it is a Φ–split Levi sub-reflection datum of 𝔾.

(3) Let 𝕄_{1} and 𝕄_{2} be two Φ–split Levi sub-reflection data containing the minimal Φ–split Levi sub-reflection datum 𝕃. Then we have

T_{W_{𝔾}}(𝕄_{1}, 𝕄_{2}) = W_{𝕄_{2}}N_{W_{𝔾}}(𝕃)W_{𝕄_{1}} .

In particular, we have

Res_{𝕄_{2}}^{𝔾} ·Ind_{𝕄_{1}}^{𝔾} = ∑_{w∈W_{𝕄_{2}} (𝕃)\W_{𝔾}(𝕃)/W_{𝕄_{1}} (𝕃)} Ind_{𝕄_{2}∩^{w}𝕄_{1}}^{𝕄_{2}} · Res_{𝕄_{2}∩^{w}𝕄_{1}}^{^{w}𝕄_{1}} ·ad(w) ,

where, for a Levi sub-reflection datum 𝕃 of 𝔾, we put W_{𝔾}(𝕃) = N_{W_{𝔾}}(𝕃)/W_{𝕃}.

Proof. For a Levi sub-reflection datum 𝕄 of 𝔾, let us denote by Z_{Φ}(𝕄) the Sylow Φ–sub-reflection datum of its center. Then 𝕄 is Φ–split if and only if 𝕄 = C_{𝔾}(Z_{Φ}(𝕄)).

(1) Suppose first that 𝕄_{1} ∩ 𝕄_{2} is defined. By 3.10, there exists a maximal torus 𝕋 of 𝔾 contained in both 𝕄_{1} and 𝕄_{2}. Then the reflection data Z_{Φ}(𝕄_{1}) and Z_{Φ}(𝕄_{2}) are both contained in the Sylow Φ–sub-reflection datum 𝕋_{Φ} of 𝕋. Now if 𝕊 is a Sylow Φ–sub-reflection datum of 𝔾 containing 𝕋_{Φ}, we see that 𝕄_{1} and 𝕄_{2} both contain C_{𝔾}(𝕊) hence 𝕊. Conversely, if 𝕄_{1} and 𝕄_{2} both contain a minimal Φ–split Levi sub-reflection datum, a fortiori they both contain a maximal torus 𝕋, thus 𝕄_{1} ∩𝕄_{2} is defined by 3.10.

The last assertion in (1) follows from the Sylow theorems.

(2) Let 𝕋 be a common maximal torus for both 𝕄_{1} and 𝕄_{2}. Then the sub-reflection datum Z_{Φ}(𝕄_{1}).Z_{Φ}(𝕄_{2}) of 𝕋 is defined in an obvious way, and it is then clear that 𝕄_{1} ∩ 𝕄_{2} = C_{𝔾}(Z_{Φ}(𝕄_{1}).Z_{Φ}(𝕄_{2})).

(3) Set 𝕊 = Z_{Φ}(𝕃) (𝕊 is a Sylow Φ–sub-reflection datum of 𝔾). Assume that w ∈ S_{W_{𝔾}}(𝕄_{1}, 𝕄_{2}). Then by what precedes, there exists a Sylow Φ–sub-reflection datum 𝕊′ of 𝔾 such that 𝕊′ is contained in both ^{w}𝕄_{1} and 𝕄_{2}. By Sylow theorems, there exist w_{1} ∈ W_{𝕄_{1}} and w_{2} ∈ W_{𝕄_{2}} such that ^{w^{−1}}𝕊′ = ^{w_{1}}𝕊 and 𝕊′ = ^{w_{2}}𝕊, from which it follows that w_{2}^{−1}ww_{1} ∈ N_{W_{𝔾}}(𝕊). The assertion now follows from the fact that N_{W_{𝔾}}(𝕊) = N_{W_{𝔾}}(𝕃). □

5B. Regular elements and Sylow theorems.

Let 𝔾 = (V, W ϕ) be a reflection datum on K. Let K′ be an extension of K which splits 𝔾. We set V′ := K′ ⊗ V. Following Springer ([Sp]), we say that

• an element v ∈ V′ is regular for W if it is fixed by no non trivial element of W, or equivalently (see 1.1) if it is fixed by no pseudo–reflection of W.

• an element wϕ in the coset W ϕ is regular if it has a regular eigenvector.

Moreover, if wϕ has a regular eigenvector for the eigenvalue ζ, we say that wϕ is ζ– regular. If wϕ is ζ–regular, it is also ζ′–regular for any root ζ′ of the minimal polynomial Φ of ζ over K. We then say that wϕ is Φ–regular.

For wϕ ∈ W ϕ and ζ a root of unity in K′, we denote by V′(wϕ, ζ) the eigenspace of wϕ corresponding to the eigenvalue ζ.

5.8. Proposition. Let ζ be a root of unity in K′ and let Φ be its minimal polynomial over K. Let 𝔾 be a reflection datum on K, and let Φ^{a(Φ)} be the largest power of Φ which divides |𝔾|. For wϕ ∈ W ϕ, we set 𝕊(wϕ, Φ) := (ker(Φ(wϕ)), (wϕ)_{|_{ker(Φ(wϕ))}}) (a Φ–subreflection datum of 𝔾). Let 𝕃(wϕ, Φ) denote its centralizer. The following assertions are equivalent.

(i) wϕ is ζ–regular, (ii) C_{W} (V′(wϕ, ζ)) = {1}, (iii) 𝕃(wϕ, Φ) is a torus, (iv) for w′ϕ ∈ W ϕ, Φ^{a(Φ)} divides the characteristic polynomial of w′ϕ if and only if

w′ϕ is W –conjugate to wϕ.

Proof of 5.8.

It is clear that (i) implies (ii). Reciprocally, if (ii) holds, it follows from 1.1 that no pseudo–reflection of W is trivial on V′(wϕ, ζ), and since the union of the fixed points of the pseudo–reflections do not cover V′(wϕ, ζ), there is a regular vector in V′(wϕ, ζ).

The assertions (ii) and (iii) are equivalent since the reflection group of 𝕃(wϕ, Φ) is C_{W} (V′(wϕ, ζ)).

The implication (i) ⇒ (iv) is due to Springer ([Sp], 6.4). Reciprocally, assume that (iv) holds. We see that any two elements of C_{W} (V (wϕ, Φ)) are W –conjugate, so have the same characteristic polynomial in their action on V. The following lemma proves then that 𝕃(wϕ, Φ) is a torus, so that (iii) holds.

5.9. Lemma. If a reflection datum 𝔾 = (V, W ϕ) is not a torus (i.e., if W ≠ 1), it has two maximal tori with different order (in other words, there are two elements wϕ and w′ϕ in W ϕ with different characteristic polynomials).

Proof of 5.9. Let |𝔾| = x^{N(𝔾)} ∏_{ζ}(x − ζ)^{a(ζ)} be the factorization of the polynomial order |𝔾| over K′[x]. By Sylow theorems (5.1) applied to the reflection datum defined over K′, for each root of unity ζ there is wϕ ∈ W ϕ whose characteristic polynomial is divisible by (x−ζ)^{a(ζ)}. Hence it suffices to check that, if W is not trivial, we have ∑_{ζ} a(ζ) > dim(V). Indeed, we have ∑_{ζ} a(ζ) = deg |𝔾| − N(𝔾) , i.e., by 3.6, (2), ∑_{ζ} a(ζ) = N^{∨}(𝔾) + r . □

□

We say that a cyclotomic polynomial Φ on K is regular for 𝔾 if there is a Φ–regular element in W ϕ.

5.10. Corollary. Let Φ be a cyclotomic polynomial on K. Let 𝔾 be a reflection datum on K, and let Φ^{a(Φ)} be the largest power of Φ which divides |𝔾|. The following assertions are equivalent :

(i) Φ is regular for 𝔾, (ii) the centralizer of a Sylow Φ–sub-reflection datum of 𝔾 is a torus, (iii) there is only one conjugacy class under W of maximal tori of 𝔾 whose order is

divisible by Φ^{a(Φ)}.

Proof of 5.10. It is an easy consequence of 5.8, and of the fact (see proof of 5.1 above) that any Sylow Φ–sub-reflection datum is of the shape 𝕊(wϕ, Φ). □

Degrees and regular elements.

Let α be a class function on 𝔾. Let wϕ be Φ–regular, so that the associated maximal toric sub-reflection datum 𝕋_{wϕ} of 𝔾 is a Sylow Φ–sub-reflection datum by 5.8. Then the congruence given in 5.4 becomes

Deg(R_{α}) ≡ α(wϕ) mod Φ ,

which can be reformulated into the following proposition.

5.11. Proposition. If wϕ is ζ–regular for some root of unity ζ, for any α ∈ CF_{uf}(𝔾) we have

Deg(R_{α})(ζ) = α(wϕ) .

Comment. The content of this section is a generalization to reflection data of the results of [BrMa1] and [BMM1] dealing with d-split Levi subgroups of reductive groups (when K = ℚ, a cyclotomic polynomial Φ is determined by the order d as roots of unity of its roots).

6. Spetsial and cyclotomic specializations of Hecke algebras

6A. Cyclotomic Hecke algebras.

Let us start with some definition and notation which will be justified later on.

Let W be a complex reflection group, with field of definition K.

Let d be an integer, a divisor of the order |ZW | of the center ZW of W. Let ζ be a root of the unity in K, of order d.

Let y be an indeterminate. We set

x := ζy^{|μ(K)|^{2}} .

6.1. Definition. A ζ–cyclotomic specialization of the set of generic indeterminates t is a morphism of ℤ_{K}–algebras

ϕ: ℤ_{K}[t, t^{−1}] → ℤ_{K}[y, y^{−1}]

with the following properties :

(a) ϕ: t_{C,j} ↦ y^{n_{C,j}} , with n_{C,j} ∈ ℤ for all C and all j .

(b) For all C ∈ A/W, the polynomial in t

∏_{j=0}^{j=e_{C}−1} (t − ζ_{e_{C}}^{j}y^{n_{C,j}})

is invariant under the action of Gal(K(y)/K(x)), i.e., belongs to ℤ_{K}[x, x^{−1}][t].

We shall often write m_{C,j} := n_{C,j}/|μ(K)| and then ϕ: u_{C,j} ↦ ζ_{e_{C}}^{j}(ζ^{−1}x)^{m_{C,j}} .

From now on, we assume that assumption 2 holds.

The corresponding cyclotomic Hecke algebra is the ℤ_{K}[x, x^{−1}]–algebra H(W, ϕ(u)) obtained as the specialization of the algebra H(W, u) through the specialization ϕ, endowed with the symmetrizing form specialized from the form t_{u}.

If there is no ambiguity about ϕ, the corresponding cyclotomic Hecke algebra is denoted by H(W, x), and its chosen symmetrizing form is denoted by t_{x}.

The algebra H(W, x) is an image of the group algebra ℤ_{K}[x, x^{−1}]B through a morphism which we still denote by ϕ: ℤ_{K}[x, x^{−1}]B → H(W, x) , and for x = ζ it specializes onto the group algebra ℤ_{K}W^{op} (while its chosen symmetrizing form specializes onto the canonical symmetric form on ℤ_{K}W^{op}).

Linear characters of the cyclotomic Hecke algebra.

Let H(W, x) be a ζ–cyclotomic Hecke algebra. Let

θ: H(W, x) ⟶ ℤ_{K}[y^{|μ(K)|}, y^{−|μ(K)|}]

be a linear character of H(W, x) (recall that y^{|μ(K)|^{2}} = ζ^{−1}x ). We denote by j_{θ} := (j_{C})_{C∈A/W} the family of integers (0 ≤ j_{C} < e_{C}) such that θ(s_{C}) = ζ_{e_{C}}^{j_{C}}(ζ^{−1}x)^{m_{C,j_{C}}} .

We set

D_{θ} := ∑_{C∈A/W} m_{C,j_{C}}N_{C}e_{C} .

Let us denote by \overline{θ} the linear character of W obtained through the specialization x ↦ ζ. Thus we have

\overline{θ}(s_{C}) = ζ_{e_{C}}^{j_{C}} .

6.2. Lemma. Let w be an element of B such that w^{d} = π, and let w be its image in W. We have

θ(w) = \overline{θ}(w)(ζ^{−1}x)^{D_{θ}/d} .

Proof of 6.2. This is a particular case of 1.22 (the reader may also give a direct proof, just using the value of π in the abelianized group of B – see [BMR1], 2.25). □

Principal cyclotomic Hecke algebras.

6.3. Definition. A ζ–cyclotomic algebra H(W, x) is said to be principal if there exists a linear character

θ_{0} : H(W, x) ⟶ ℤ_{K}[x, x^{−1}] , s_{C} ↦ ζ_{e_{C}}^{j_{C}}(ζ^{−1}x)^{m_{C}}

such that (p1) we have m_{C} ≥ m_{C,j} for all j (0 ≤ j < e_{C}), (p2) whenever w is an element of B such that w^{d} = π, we have θ_{0}(w) = x^{D_{θ_{0}} /d} .

The character θ_{0} is then called a principal character of H(W, x).

Remark. By 6.2 above, condition (p2) above is equivalent to the condition (p2’) \overline{θ}_{0}(w) = ζ^{D_{θ_{0}} /d} . Assume now that ζ = exp(2aπi/d). Then the condition (p2) above is equivalent to

the following condition :

(p2”) d divides ∑_{C∈A/W} j_{C}N_{C} − am_{C} \frac{N_{C}e_{C}}{d} .

Indeed, this follows from the fact that θ_{0}(w) = ∏_{C∈A/W} θ_{0}(s_{C})^{N_{C}e_{C}/d} .

Examples.

(1) The spetsial Hecke algebra H(W) is the principal cyclotomic Hecke algebra associated with the 1–cyclotomic specialization

(6.4)  \{ \begin{matrix}u_{C,0} ↦ x , \\ u_{C,j} ↦ ζ_{e_{C}}^{j} for j > 0 . \end{matrix}

In other words, the images of the elements s_{H,γ} in the algebra H_{x}(W) satisfy the equations

(s_{H,γ} − x)(1 + s_{H,γ} + s_{H,γ}^{2} + ⋯ + s_{H,γ}^{e_{H}−1}) = 0 .

In particular, one sees that if W is a Coxeter group, the spetsial Hecke algebra H(W) is the ordinary generic Hecke algebra over ℤ_{K}[x, x^{−1}].

(2) Let W = G(4, 2, 2) and let ζ = i. Recall (see for example [BMR]) that the corresponding braid group B is generated by three generators of the monodromy s, s′, s′′, such that ss′s′′ = s′s′′s = s′′ss′, corresponding to reflections s, s′, s′′ of order 2 in W which satisfy the relations s′′s′s = s′ss′′ = ss′′s′.

The relations

\{ \begin{matrix}ss′s′′ = s′s′′s = s′′ss′ \\ (s − 1)(s − x^{2}) = (s′ − 1)(s′ − x^{2}) = (s′′ − 1)(s′′ − x^{2}) = 0 \end{matrix}

define a principal cyclotomic Hecke algebra associated with W.

Comment. In the case of a reductive group G over a field of characteristic p, let W be the Weyl group of G, let d be a positive integer, let (L, λ) be a d–split pair (see [BMM1]), and let 𝔘(G^{F}; (L, λ)) be the set of unipotent irreducible characters of G^{F} which occur as constituents of R_{L}^{G}(λ). The group W_{G}(L, λ) := N_{G^{F}} (L, λ)/L^{F} is a complex reflection group in an appropriate vector space (see loc. cit.).

It is conjectured (see for example [BrMa2],1C, or [Bro2], 4.2 ; see [BrMi] for some evidence on this conjecture) that there exist

• a Deligne–Lusztig variety X_{G}(L, λ) endowed with a sheaf F_{G}(L, λ),

• a ζ_{d}–cyclotomic Hecke algebra H_{G}(L, λ)(x) (where ζ_{d} = exp(2πi/d)) associated with the reflection group W_{G}(L, λ), such that, whenever ℓ is a prime number different from p,

C1. the endomorphism algebra End_{\overline{ℚ}_{ℓ}G^{F}} (H_{c}^{∗}(X_{G}(L, λ), F_{G}(L, λ))) of the compact support ℓ-adic cohomology of (X_{G}(L, λ), F_{G}(L, λ)) is isomorphic to the specialization of H_{G}(L, λ)(x) at x := q ;

C2. through this specialization, the symmetrizing form t_{G}(L, λ)(x) of H_{G}(L, λ)(x) specializes to the form t_{G}(L, λ)(q) such that

Deg(R_{L}^{G}(λ))t_{G}(L, λ)(q) = ∑_{n}(−1)^{n}tr (·; H_{c}^{n}(X_{G}(L, λ), F_{G}(L, λ))) .

The above conjecture implies the existence of a bijection

Irr(H_{G}(L, λ)) ⟶^{∼} 𝔘(G^{F}; (L, λ)) , ψ ↦ γ_{ψ}

(see loc. cit., Fundamental theorem 3.2), with the following properties, which can be checked from [BrMa2], [Ma1], [Ma2], and [Ma5] :

For all ψ ∈ Irr(H_{G}(L, λ)), its Schur element S_{ψ}(x) is a Laurent polynomial in x (and not only in a fractional power of x), and the generic degree of the corresponding unipotent character γ_{ψ} is given by the formula

ε_{ψ}Deg_{γ_{ψ}} (q) = \frac{Deg(R_{L}^{G}(1))(q)Deg(λ)(q)}{S_{ψ}(q)} ,

where ε_{ψ} = ±1.

Note that if d = 1 and λ = 1, we have W_{G}(L, λ) = W, the corresponding cyclotomic Hecke algebra H(W)(x) is the spetsial Hecke algebra H(W), and the polynomial Deg(R_{L}^{G}(1))(q)Deg(λ)(q) is the Poincaré polynomial P_{W} (q) of W. And indeed, for χ ∈ Irr(W), the generic degree of the corresponding unipotent character γ_{χ} is given by the formula

Deg_{γ_{χ}}(q) = \frac{P_{W} (q)}{S_{χ}(q)} .

Moreover, one conjectures that :

If λ = 1 (the trivial character of L^{F}), then the algebra H_{G}(L, 1)(x) is principal, the element of 𝔘(G^{F}; (L, 1)) corresponding to the principal character θ_{0} of H_{G}(L, 1)(x) is the trivial character of G^{F}, and we have

D_{θ_{0}}/d = dim(X_{G}(L, 1)) .

Our “spetses” and their unipotent characters own properties which generalize the above properties of finite reductive groups (see [BMM2]). What follows will be useful for studying generic degrees in this more general context, and to compute a priori the parameters of the cyclotomic Hecke algebras.

6B. Some definitions and notation about palindromicity.

Let P (x) ∈ ℂ(x) (or more generally suppose that P is a meromorphic function on ℂ). We say that P (x) is semi–palindromic if there exist an integer m ∈ ℤ and a norm one complex number ξ such that

(6.5)  P (1/x)^{∗} = ξx^{−m}P (x) .

Remark. The preceding definition of semi–palindromicity differs from the definition given in [Ma3]. In particular, all cyclotomic polynomials on algebraic number fields are now semi– palindromic, while the fake degrees of characters of a non real reflection group are never semi–palindromic.

The following statements are obvious to check :

(1) A constant is semi–palindromic.

(2) If P (x) and Q(x) are semi–palindromic, so is P (x)Q(x).

(3) If ζ is a norm one complex number and if P (x) is semi–palindromic, then the element P^{ζ}(x) := P (ζ^{−1}x) is semi–palindromic.

For P (x) ≠ 0, define the valuation val_{x}(P) (also denoted by a) as the order of 0 as a zero of P (x) (a < 0 if 0 is a pole of P (x)). Let c_{a} be the nonzero complex number defined by

c_{a} := (x^{−a}P (x))_{|_{x=0}} .

Define the degree deg_{x}(P) (also denoted by A) as the valuation of P (1/x). Let c_{A} be the nonzero complex number defined by

c_{A} := (x^{A}P (1/x))_{|_{x=0}} .

Thus if P (x) ∈ ℂ[x, x^{−1}] is a Laurent polynomial, we have

P (x) = c_{a}x^{a} + c_{a+1}x^{a+1} + ⋯ + c_{A−1}c^{A−1} + c_{A}x^{A}

for some coefficients c_{j} .

The following lemma is immediate.

6.6. Lemma. Let P (x) satisfy the condition 6.5 above.

(1) We have m = a + A, and c_{A}^{∗} = ξc_{a}. Moreover,

mP (x) = xP′(x) + ξ^{∗}x^{m−1}P′(1/x)^{∗} .

(2) Let ζ be a norm one complex number such that P (ζ) ≠ 0.

(a) We have

ξ = ζ^{m} \frac{P (ζ)^{∗}}{P (ζ)} and m = ζ \frac{P′(ζ)}{P (ζ)} + ζ^{∗} \frac{P′(ζ)^{∗}}{P (ζ)^{∗}} = 2ℜ (ζ \frac{P′(ζ)}{P (ζ)}) .

(b) If moreover P (ζ) ∈ ℝ, we have ξ = ζ^{m} . In particular if ζ = 1, we have then

ξ = 1 and m = \frac{P′(1) + P′(1)^{∗}}{P (1)} .

Remark. Assume that P (y) ∈ ℂ(y), and set x := y^{n} for some natural integer n. Then by abuse of notation, we define

deg_{x}(P) := \frac{deg_{y}(P)}{n} and val_{x}(P) := \frac{val_{y}(P)}{n} .

6C. On spetsial cyclotomic Hecke algebras.

We recall that in this paragraph we are assuming assumption 2 holds.

6.7. Proposition. Let us denote by H_{x}(W) the specialization of the generic Hecke algebra through 6.4 above. Then,

(1) the polynomial P_{W} (u) specializes to a Laurent polynomial P_{W} (x) ∈ ℤ_{K}[x, x^{−1}] which is palindromic and satisfies the relation

P_{W} (x) = x^{N^{∨}}P_{W} (x^{−1}) ,

(2) the absolute generic degree Deg_{χ}(t) specializes to Deg_{χ}(x) ∈ K(x^{1/|μ(K)|}) which satisfies the relation

Deg_{χ}(x) = x^{\frac{N(χ)+N(χ^{∗})}{χ(1)}} Deg_{χ}(x^{−1})^{∗} ,

(3) the character χ_{t} specializes to a character χ_{x} which satisfies the relation

ω_{χ_{x}}(π) = x^{N+N^{∨}− \frac{N(χ)+N(χ^{∗})}{χ(1)}} .

Proof of 6.7.

(1) By 2.12 and 2.13 above, it suffices to check that M_{W} (u) specializes onto x^{N^{∨}}. This results from the following lemma.

50 Michel Broué, Gunter Malle, Jean Michel

6.8. Lemma. Under 6.4, (a) ω_{1_{t}}(π) specializes onto x^{N+N^{∨}}.

(b) t_{u}(π) specializes onto x^{N}. Proof of 6.8. The first assertion is obvious. Let us check the second one. We have

t_{u}(π) = (−1)^{N^{∨}} ∏_{{H∈A \atop 0≤j≤e_{H} −1}} u_{H,j} = (−1)^{N^{∨}} ∏_{{C∈A/W \atop 0≤j≤e_{C} −1}} u_{C,j}^{N_{C}}

hence t_{u}(π) specializes to

(−1)^{N^{∨}} ∏_{C∈A/W} x^{N_{C}} ∏_{{C∈A/W \atop 1≤j≤e_{C} −1}} exp (\frac{2πijN_{C}}{e_{C}} )

= (−1)^{N^{∨}}x^{∑_{C∈A/W} N_{C}} exp ( ∑_{C∈A/W} \frac{2πiN_{C}}{e_{C}} ∑_{1≤j<e_{C}} j)

= (−1)^{N^{∨}}x^{N} exp (πi ∑_{C∈A/W} N_{C}(e_{C} − 1))

= (−1)^{N^{∨}}x^{N}(−1)^{∑_{C∈A/W} N_{C}(e_{C}−1)}

= x^{N}

(by 1.2). □

(2) It suffices to compute the specialization M_{χ}(x) of M_{χ}(u). By 2.12, we have

M_{χ}(x) = exp (\frac{2iπ}{χ(1)}(N(χ) − ∑_{(C,j)} jN_{C}m_{C,j}^{χ})) x^{∑_{C}(N_{C}e_{C}−^{\frac{N_{C} e_{C} m{χ \atop C,0}}{χ(1)}} )} .

By the equalities

∑_{(C,j)} jN_{C}m_{C,j}^{χ} = N(χ) ,

and

∑_{C∈A/W} (N_{C}e_{C} − \frac{N_{C}e_{C}m_{C,0}^{χ}}{χ(1)} ) = \frac{N(χ) + N(χ^{∗})}{χ(1)} ,

(cf. 1.11 above), we see that

M_{χ}(x) = x^{\frac{N(χ)+N(χ^{∗})}{χ(1)}} .

(3) Since

ω_{χ_{t}}(π) = ω_{1_{t}}(π)/M_{χ}(u) ,

the formula

ω_{χ_{x}}(π) = x^{N+N^{∨}− \frac{N(χ)+N(χ^{∗})}{χ(1)}} ,

results from what precedes. □

6.9. Corollary. The sum of the valuation a_{χ} and of the degree A_{χ} of the generic degree Deg_{χ}(x) of χ is computable from its fake degree Feg_{χ}(x) : we have

a_{χ} + A_{χ} = \frac{N(χ) + N(χ^{∗})}{χ(1)} .

Proof of 6.9. It results from 6.7, (2) and 6.6, (1). □

6D. Palindromicity, fake degrees, generic degrees.

By [Ma3], 4.8, we know that the field K(x^{1/|μ(K)|}) contains the values of the absolutely irreducible characters of the spetsial cyclotomic Hecke algebra H_{x}(W). Following loc. cit., we denote by δ the generator of Gal(K(x^{1/|μ(K)|})/K(x)) defined by

δ(x^{1/|μ(K)|}) = exp (2πi/|μ(K)|) x^{1/|μ(K)|} .

We denote by ι the automorphism of order 2 of K(x^{1/|μ(K)|}) defined as the composition of the complex conjugation by the automorphism δ, i.e.,

ι(λ) := δ(λ^{∗}) for λ ∈ K(x^{1/|μ(K)|}) .

We have an operation of Gal(K(x^{1/|μ(K)|})/K(x)) on the set Irr(W) of irreducible characters of W (see also above §1) defined as follows :

g(χ)_{x} = g(χ_{x})

(for g ∈ Gal(K(x^{1/|μ(K)|})/K(x)) and χ ∈ Irr(W)).

The following property is proved in loc. cit., (6.5), through a case by case analysis. Note that Opdam [Op1] has given a general proof for the case where W is a Coxeter

group, and that his proof can be generalized to any complex reflection group W (see [Op2]) provided one assumes that presentations like in [BMR] hold for B (see loc. cit., §6).

6.10. Proposition. Assume assumption 1 holds. Whenever χ ∈ Irr(W), we have

Feg_{χ}(x) = x^{m} Feg_{ι(χ)}(1/x)

for some natural integer m.

It follows from this assertion that the polynomial P_{χ}(x) := Feg_{χ}(x) Feg_{ι(χ)}(x) satisfies the relation

P_{χ}(x) = x^{2m}P_{χ}(1/x) ,

hence is palindromic. Since P_{χ}(1) = χ(1)^{2} is non zero, it follows from lemma 6.6, (2)(b) above (see also loc. cit.) that

2m = 2\frac{P′_{χ}(1)}{P_{χ}(1)} ,

from which it is immediate that

m = \frac{N(χ) + N(ι(χ))}{χ(1)} .

By 1.24 above, we know that N(ι(χ)) = N(χ^{∗}) , which implies

(6.11)  m = \frac{N(χ) + N(χ^{∗})}{χ(1)} .

Let us reformulate what precedes by introducing the valuation b_{χ} and the degree B_{χ} of Feg_{χ}(x), and by using [Ma3], 6.5. Since the valuation and the degree of the polynomial Feg_{χ}(x) Feg_{ι(χ)}(x) are respectively b_{χ} + b_{ι(χ)} and B_{χ} + B_{ι(χ)} we have

(6.12)  \frac{(b_{χ} + B_{χ}) + (b_{ι(χ)} + B_{ι(χ)})}{2} = N^{∨} − ∑_{ρ∈Ref(W)} \frac{χ(ρ)}{χ(1)} = \frac{N(χ) + N(χ^{∗})}{χ(1)} .

It results from 6.7 and 6.9 above that Deg_{χ}(x) is semi–palindromic, and that if a_{χ} and A_{χ} are respectively its valuation and degree in x, we have

a_{χ} + A_{χ} = \frac{N(χ) + N(χ^{∗})}{χ(1)} .

Then we deduce from 6.12 the supplementary equality

(6.13)  \frac{(b_{χ} + B_{χ}) + (b_{ι(χ)} + B_{ι(χ)})}{2} = a_{χ} + A_{χ} .

6E. On generic degrees of cyclotomic Hecke algebras.

Let ζ be a root of unity of order d. Let H(W, x) be a ζ–cyclotomic Hecke algebra defined by a cyclotomic specialization ϕ: t_{C,j} ↦ y^{n_{C,j}} where y^{|μ(K)|^{2}} = ζ^{−1}x.

Let Irr(H(W, x)) denote the set of (absolutely) irreducible characters of the algebra K(y)H(W, x).

Let P (y) ∈ K(y) be semi–palindromic. Let ψ ∈ Irr(H(W, x)).

• We denote by \overline{ψ} the character of W defined by ψ through the specialization y ↦ 1. Thus in particular we have \overline{ψ}(s_{C}) = (ψ(s_{C}))_{y=1}.

• The P –generic degree of ψ is the element Deg_{ψ}^{(P)}(y) ∈ K(y) defined as follows :

Deg_{ψ}^{(P)}(y) := \frac{P (y)}{S_{ψ}(y)} ,

where we denote by S_{ψ}(y) the Schur element of ψ relatively to the form t_{x}.

Comment. In the applications, we shall choose P (y) = Deg(R_{L}^{G}(1))(x)Deg(λ)(x) (see comment in §6.A above).

• We denote by a_{ψ} and A_{ψ} respectively the (generalized) valuation and degree of Deg_{ψ}^{(P)}(y) in x.

Note that S_{ψ}(y) is semi–palindromic, since by 2.8 it satifies the equality

(6.14)  S_{χ}(y^{−1})^{∗} = \frac{t_{x}(π)}{ω_{ψ}(π)}S_{χ}(y) .

It follows that Deg_{ψ}^{(P)}(y) is semi–palindromic, so satisfies the identity

Deg_{ψ}^{(P)}(y^{−1})^{∗} = ξ_{ψ}x^{−(a_{ψ}+A_{ψ})}Deg_{ψ}^{(P)}(y)

for some root of the unity ξ_{ψ}.

In what follows, we choose a linear character θ_{0} of H(W, x), such that θ_{0}(s_{C}) = ζ_{e_{C}}^{j_{C}}(ζ^{−1}x)^{m_{C}} . We set

\{ \begin{matrix}D_{0} := D_{θ_{0}} = ∑_{C∈A/W} m_{C}N_{C}e_{C} , \\ a_{0} := a_{θ_{0}} and A_{0} := A_{θ_{0}} . \end{matrix}

6.15. Proposition. Let w be an element of B such that w^{d} = π.

(1) We have

ω_{ψ}(w) = ω\overline{ψ}(w)(ζ^{−1}x)^{\frac{D_{0}−((a_{ψ} +A_{ψ})−(a_{0}+A_{0}))}{d}} .

(2) Assume that H(W, x) is principal, and that θ_{0} is its principal character. Assume moreover that Deg_{θ_{0}}^{(P)}(y) = 1. We have

ω_{ψ}(w) = ω\overline{ψ}(w)(ζ^{−1}x)^{\frac{D_{0}−(a_{ψ} +A_{ψ})}{d}} .

Proof of 6.15. The second assertion is a particular case of the first one. Let us prove (1).

Let us set

F_{ψ}(y) := \frac{Deg_{ψ}^{(P)}(y)}{Deg_{θ_{0}}^{(P)}(y)} .

• On one hand, F_{ψ}(y) is semi–palindromic (as a quotient of two semi–palindromic functions) and we have, up to multiplication by a root of unity,

F_{ψ}(y^{−1})^{∗} = x^{−((a_{ψ}+A_{ψ})−(a_{0}+A_{0}))}F_{ψ}(y) .

• On the other hand, since F_{ψ}(y) = \frac{S_{θ_{0}}(y)}{S_{ψ}(y)} , it follows from 6.14 that

F_{ψ}(y^{−1})^{∗} = \frac{ω_{ψ}(π)}{θ_{0}(π)}F_{ψ}(y) .

Thus we have (up to multiplication by a root of unity)

ω_{ψ}(π) = θ_{0}(π)x^{−((a_{ψ}+A_{ψ})−(a_{0}+A_{0}))} = x^{D_{0}−((a_{ψ}+A_{ψ})−(a_{0}+A_{0}))} ,

hence (up to multiplication by a root of unity)

ω_{ψ}(w) = x^{\frac{D_{0}−((a_{ψ} +A_{ψ})−(a_{0}+A_{0}))}{d}} .

Specializing through the map y ↦ 1 gives then the first assertion of 6.15. □

The preceding assertion allows us to generalize some of the results of [BrMi] (see loc. cit., 5.12).

For C ∈ A/W and 0 ≤ j < e_{C}, we denote by θ_{C,j} the linear character of H(W, x) defined by

θ_{C,j}(s_{C′}) = \{ \begin{matrix}ζ_{e_{C}}^{j}(ζ^{−1}x)^{m_{C,j}} if C′ = C , \\ θ_{0}(s_{C′}) if C′ ≠ C . \end{matrix}

We set

Deg_{C,j}^{(P)}(y) := Deg_{θ_{C,j}}^{(P)}(y) ,
ω_{C,j}(y) := ω_{θ_{C,j}}(y) ,
a_{C,j} := a_{θ_{C,j}} and A_{C,j} := A_{θ_{C,j}} .

• On one hand, from the definition of θ_{C,j}, it is clear that

ω_{C,j}(π) = ω_{θ_{0}}(π) (\frac{ζ_{e_{C}}^{j}(ζ^{−1}x)^{m_{C,j}}}{ζ_{e_{C}}^{j_{C}}(ζ^{−1}x)^{m_{C}}} )^{N_{C}e_{C}} .

Since ω_{θ_{0}}(π) is a monomial of degree D_{0} in x, we see that ω_{C,j}(π) is a monomial of degree D_{0} + (m_{C,j} − m_{C})N_{C}e_{C}.

• On the other hand, it results from that ω_{C,j}(π) is a monomial of degree D_{0} − ((a_{C,j} + A_{C,j}) − (a_{0} + A_{0})).

The following statement is now obvious.

6.16. Proposition.

(1)

(m_{C} − m_{C,j})N_{C}e_{C} = (a_{C,j} + A_{C,j}) − (a_{0} + A_{0}) .

(2) In particular, if H(W, x) is principal, if θ_{0} is the principal character and if Deg_{θ_{0}}^{(P)} = 1, we have

(m_{C} − m_{C,j})N_{C}e_{C} = (a_{C,j} + A_{C,j}) .

Appendix : Symmetric algebras

For the convenience of the reader, we collect and prove here various results about specializations of algebras and about symmetric algebras. Most of these results are known. The presentation given here is inspired by various ideas in [Bro1], [Ge], [GeRo] and [Ro].

In what follows the following notation and assumptions will be in force :

• We denote by R an integrally closed commutative noetherian domain, with field of fractions denoted by F.

• We denote by H an R–algebra which is a free R–module of finite rank.

[Actually, for most of the results proved below it would be enough to assume that H is a finitely generated projective R–module.]

• The R–dual of the module H is H^{∨} := Hom(H, R) . The map

H^{∨} ⊗ H → End_{R}(H) , ϕ ⊗ h ↦ (h′ ↦ ϕ(h′)h)

is an isomorphism. The trace tr_{H}: End_{R}(H) → R is defined, through the preceding isomorphism, by the natural pairing H^{∨} ⊗ H → R . We still denote by

tr_{H}: H → R

the composition of the trace with the natural monomorphism H ↪ End_{R}(H) defined by the left regular representation of H.

• We set F H := F ⊗_{R} H . More generally, if R → R′ is a ring morphism, we set R′H := R′ ⊗_{R} H .

Values of characters.

7.1. Proposition. Let χ be the character of some finite dimensional F H–module. (1) For all h ∈ H, we have χ(h) ∈ R.

(2) Assume moreover that R is a ℤ–graded ring, and that H is a graded R–module which has an R–basis consisting of homogeneous elements. Then, for all h ∈ H with degree n, χ(h) is an element with degree n of R.

Proof of 7.1.

(1) For h ∈ H, the characteristic polynomial Cha(h)(x) of the left multiplication by h in H is a unitary element of R[x]. Hence its roots are all integral over R.

In order to prove 7.1, we may assume that χ is the character of an irreducible F H– module. In this case, the characteristic polynomial Cha_{χ}(h)(x) of h acting on this module divides Cha(h)(x) and is unitary, from which it follows that its coefficients are integral over R. Since R is integrally closed, we see that Cha_{χ}(h)(x) ∈ R[x]. In particular, we have χ(h) ∈ R.

(2) Keeping the same notation as above, we see now that, if

Cha(h)(x) = x^{d} − a_{d−1}(h)x^{d−1} + ⋯ + (−1)^{d}a_{0}(h) ,

then for each j the coefficient a_{j}(h) is a homogeneous element of degree n(d − j) in R (since the matrix of the left multiplication by h on a basis consisting of homogeneous elements must have homogeneous entries all of degree n).

Let us define a graduation on R[x] by assigning the degree ℓ + nm to λx^{m}, for λ homogeneous of degree ℓ. Then we see that Cha(h)(x) is a homogeneous element of degree dn in R[x].

Any divisor of a homogeneous element is again homogeneous. So Cha_{χ}(h)(x) is homogeneous, and since its degree as a polynomial in x is χ(1), it is homogeneous of degree nχ(1). This proves in particular that the coefficient of x^{χ(1)−1}, namely χ(h), is homogeneous of degree n. □

Let p: R ↠ k be a morphism from R onto a field k. For φ: H → R a linear form, we denote by φ_{k}: kH → k the function such that the following diagram is commutative :

H ⟶^{φ} R
p↓  ^{p}↓
kH ⟶^{φ_{k}} k .

In particular, by 7.1 above, any character χ of a F H–module defines a central function χ_{k}: kH → k.

Under certain hypotheses, the function χ_{k} is again a character, as shown in particular by the following classical result, known as “Tits deformation theorem”.

7.2. Theorem. Assume that F H and kH are split semi–simple. Then the map χ ↦ χ_{k} defines a bijection from the set Irr(F H) of irreducible characters of F H onto the set Irr(kH) of irreducible characters of kH, which preserves the degrees of characters.

Proof of 7.2. We only sketch a proof of this well known result.

Let n be the rank of H over R. We extend the scalars to the polynomial algebra R[x_{1}, x_{2}, …, x_{n}] to get the algebra R[x_{1}, x_{2}, …, x_{n}]H. Choose a basis (h_{1}, h_{2}, …, h_{n}) of H over R and consider the generic element g := ∑_{j=1}^{j=n} x_{j}h_{j}. Let Cha(g)(x) be the characteristic polynomial of the left multiplication by g. Then we have the following factorization in F (x_{1}, x_{2}, …, x_{n})[x] (hence in R[x_{1}, x_{2}, …, x_{n}][x] since R[x_{1}, x_{2}, …, x_{n}][x] is integrally closed) :

(*)  Cha(g)(x) = ∏_{χ∈Irr(F H)} Cha_{χ}(g)(x)^{d_{χ}} ,

where Cha_{χ}(g)(x) is the characteristic polynomial of g in a representation with character χ and where d_{χ} is the degree of χ, i.e., the degree (in x) of Cha_{χ}(g)(x). It is easy to see that each Cha_{χ}(g)(x) is irreducible.

Applying the morphism p, we get a decomposition of p(Cha(g))(x) in the algebra k[x_{1}, x_{2}, …, x_{n}][x] :

p(Cha(g))(x) = ∏_{χ∈Irr(F H)} p(Cha_{χ}(g))(x)^{d_{χ}} .

On the other hand, since p(Cha(g))(x) is the characteristic polynomial of the left multiplication by g in k[x_{1}, x_{2}, …, x_{n}]H, this polynomial has a decomposition into irreducible factors in k[x_{1}, x_{2}, …, x_{n}][x] which is analogous to (*) above :

p(Cha(g))(x) = ∏_{ϕ∈Irr(kH)} Cha_{ϕ}(g)(x)^{d_{ϕ}} ,

where d_{ϕ} is the degree of Cha_{ϕ}(g)(x). It is easy to see that the decompositions must coincide, whence the theorem 7.2 follows. □

Symmetric algebras.

7.3. Definitions.

• A linear form t: H → R is called central if t(hh′) = t(h′h) for all h, h′ ∈ H.

• A symmetrizing form on H is a central linear form t: H → R such that the morphism

t̂: H → H^{∨} , h ↦ (h′ ↦ t(hh′))

is an isomorphism.

• The algebra H is said to be symmetric if there exists a symmetrizing form.

• If the trace form tr_{H}: H → R is symmetrizing, we say that H is trace symmetric.

Let us note some elementary facts about specializations and symmetrizing forms.

7.4. Proposition. Let p: R ↠ k be a morphism from R onto a field k. Assume that there is a central form t: H → R which defines a symmetrizing form on kH. Then t is a symmetrizing form for F H.

In particular, if kH is trace symmetric, then F H is trace symmetric.

Proof of 7.4. Let 𝔪 be the kernel of p. We set \overline{t} := p · t: H → k. The discriminant of the form \overline{t} on an R–basis of H does not belong to 𝔪, since its image through p has to be non zero. This proves that t is a symmetrizing form for the algebra R_{𝔪}H (where R_{𝔪} is the localization of R at 𝔪), hence in particular is a symmetrizing form for F H. □

Symmetrizing forms and separability.

If the algebra F H is trace symmetric, then it is separable, i.e., whenever L is an extension of F, the algebra LH is semi–simple. This is a particular case of a more general result which we prove below (see 7.7).

Let us first introduce the notion of “pseudo–character” (see for example [Ro]).

Whenever τ: H → R is a central function on H, for all natural integers n we construct a symmetric function S_{n}(τ): H^{n} → R by the formula

S_{n}(τ) := ∑_{σ∈𝔖_{n}} ε(σ)(σ · τ) ,

where, for σ = σ_{1}σ_{2} ⋯ σ_{r} the decomposition of σ into a product of disjoint cycles, and σ_{j} = (j_{1}, …, j_{m}), we set

(σ · τ)(x_{1}, x_{2}, …, x_{n}) := τ(σ_{1}(x))τ(σ_{2}(x)) ⋯ τ(σ_{r}(x))

where

σ_{j}(x) := x_{j_{1}} ⋯ x_{j_{m}} .

We say that the function τ is a pseudo–character of H if there exists an integer d such that S_{d}(τ) = 0.

The following result (see [Ro], 3.1) justifies the terminology.

7.5. Lemma. Any character of a F H–module is a pseudo–character.

Brief indications about the proof of 7.5. One reduces easily to the case where H is a matrix algebra over a characteristic zero field. Then the assertion becomes a formal property of traces of matrices. □

Pseudo–characters inherit from characters the property of vanishing on nilpotent elements (see [Ro], 2.6).

7.6. Lemma. Any pseudo–character vanishes on nilpotent elements of H.

7.7. Proposition. Assume that the algebra F H has a symmetrizing form which is a linear combination of pseudo–characters. Then F H is separable.

Proof of 7.7. Let t be a symmetrizing form on F H which is a linear combination of pseudo–characters. Let L be an extension of F. Since the elements of the Jacobson radical of LH are nilpotent, they are contained in the kernel of all pseudo–characters, hence in the kernel of the map t̂, which proves that the Jacobson radical of LH equals {0}. □

More on symmetric algebras.

From now on we assume that

H is endowed with a symmetrizing form t.

The isomorphism H^{∨}⊗H ⟶^{∼} End_{R}(H) , together with the isomorphism t̂: H ⟶^{∼} H^{∨} , defines an isomorphism H ⊗ H ⟶^{∼} End_{R}(H) .

We denote by T_{H} the element of H⊗H corresponding to the identity on H through the preceding isomorphism. Writing T_{H} = ∑_{j} h′_{j} ⊗h_{j} , we get the following characterization of T_{H} :

(7.8)  h = ∑_{j} t(hh′_{j})h_{j} for all h ∈ H .

Remark. If (e_{j})_{j} is an R–basis of H, and if (e′_{j})_{j} is the dual basis (defined by the condition t(e_{j}e′_{j′}) = δ_{j,j′}), then we have T_{H} = ∑_{j} e′_{j} ⊗ e_{j} .

The following properties of the element T_{H} are straightforward.

7.9. Lemma.

(1) For all h ∈ H, we have ∑_{j} hh′_{j} ⊗ h_{j} = ∑_{j} h_{j} ⊗ h′_{j}h .

(2) We have ∑_{j} h′_{j} ⊗ h_{j} = ∑_{j} h_{j} ⊗ h′_{j} , and hT_{H} = T_{H}h for all h ∈ H .

(3) For all h ∈ H, we have

h = ∑_{j} t(hh′_{j})h_{j} = ∑_{j} t(hh_{j})h′_{j} = ∑_{j} t(h′_{j})h_{j}h = ∑_{j} t(h_{j})h′_{j}h .

Proof of 7.9.

Let us prove (1). By 7.8, we have hh′_{j} = ∑_{i} t(hh′_{j}h′_{i})h_{i} , hence

∑_{j} hh′_{j} ⊗ h_{j} = ∑_{i,j} t(hh′_{j}h′_{i})h_{i} ⊗ h_{j}

= ∑_{i} h_{i} ⊗ ∑_{j} t(h′_{i}hh′_{j})h_{j}

= ∑_{i} h_{i} ⊗ h′_{i}h .

Assertions (2) and (3) follow immediately from (1). □

For τ: H → R a linear form, we denote by τ^{∨} the element of H defined by the condition

t(τ^{∨}h) = τ(h) for all h ∈ H .

It is easy to check the following set of properties.

7.10. Lemma.

(1) τ is central if and only if τ^{∨} is central in H.

(2) We have τ^{∨} = ∑_{j} τ(h′_{j})h_{j} = ∑_{j} τ(h_{j})h′_{j} , and more generally, for all h ∈ H, we have τ^{∨}h = ∑_{j} τ(h′_{j}h)h_{j} = ∑_{j} τ(h_{j}h)h′_{j} .

Let χ be the character of an absolutely irreducible F H–module V_{χ}. We denote by

ρ_{χ}: F H ↠ End_{F} (V_{χ})

the natural (epi)morphism. It restricts to a morphism

ω_{χ}: ZH → R

(where we identify R to a subring of the center of End_{F} (V)).

The Schur element of χ is by definition the element of R defined by

(7.11)  S_{χ} := ω_{χ}(χ^{∨}) .

Notice that by 7.10, (2), we have

S_{χ}χ(1) = ∑_{j} χ(h′_{j})χ(h_{j}) .

7.12. Proposition. Assume that F H is split semi–simple.

(1) For each irreducible character χ of F H, let e_{χ} be the primitive idempotent of the center ZF H of the algebra F H associated with χ. Then we have

χ^{∨} = S_{χ}e_{χ} .

(2) We have

t = ∑_{χ∈Irr(F H)} \frac{1}{S_{χ}} χ .

Proof of 7.12.

(1) Since, for all h ∈ H, we have χ(e_{χ}h) = χ(h) , we see that t(χ^{∨}e_{χ}h) = t(χ^{∨}h) , which proves that χ^{∨} = χ^{∨}e_{χ} . The desired equality results from the fact that, for all z ∈ ZF H, we have z = ∑_{χ∈Irr(F H)} ω_{χ}(z)e_{χ} .

(2) Through the isomorphism between H and its dual, the equality

t = ∑_{χ∈Irr(F H)} \frac{1}{S_{χ}} χ

is equivalent to

1 = ∑_{χ∈Irr(F H)} \frac{1}{S_{χ}} χ^{∨} ,

which is obvious by (1) above. □

7.13. Theorem. Assume that H is endowed with a symmetrizing form t, and that F H is split semi-simple. For each χ ∈ Irr(F H), let us denote by S_{χ} the corresponding Schur element.

Let p: R ↠ k be a morphism from R onto a field k. Assume that kH is split. Then the following assertions are equivalent :

(i) For all χ ∈ Irr(F H), we have p(S_{χ}) ≠ 0.

(ii) kH is (split) semi–simple.

Proof of 7.13. Notice that the form t defines a symmetrizing form t_{k} on kH. (i) ⇒ (ii) : We have

t_{k} = ∑_{χ∈Irr(F H)} \frac{1}{p(S_{χ})}χ_{k} .

In particular, we see that t_{k} is a linear combination of pseudo–characters of kH, hence kH is separable by 7.7 above.

(ii) ⇒ (i) : By 7.2, we know that Irr(kH) = {χ_{k} | χ ∈ Irr(F H)}. It is then clear that S_{χ_{k}} = p(S_{χ}), which proves that p(S_{χ}) ≠ 0. □

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Michel Broué, Institut Universitaire de France, Université Paris 7 Denis-Diderot, et UMR 7586 du CNRS, Case 7012, 2 Place Jussieu, F–75251 Paris Cedex 05, France

   E-mail address: broue@math.jussieu.fr
   URL: http://www.math.jussieu.fr/ broue

Gunter Malle, Fachbereich Mathematik/Informatik, Universität Kassel, Heinrich– Plett–Straße 40, D–34132 Kassel, Germany

   E-mail address: malle@mathematik.uni-kassel.de

Jean Michel, UMR 7586 du CNRS, Université Paris 7 Denis–Diderot, Case 7012, 2 Place Jussieu, F–75251 Paris Cedex 05, France, et LAMFA, UPRES–A 6119 du CNRS, 33 rue Saint–Leu, F–80039 Amiens Cédex 01, France

   E-mail address: jmichel@math.jussieu.fr
   URL: http://www.math.jussieu.fr/ jmichel