Generalized Harish-Chandra Theory

Michel Broué & Gunter Malle

Université Paris 7 Denis–Diderot et Institut Universitaire de France, Case 7012, 2 Place Jussieu, F–75251 Paris Cedex 05, France

IWR, Universität Heidelberg, 69120 Heidelberg, Germany

1 Introduction

In the study of subgroup structure, ordinary and modular representation theory of finite groups of Lie type it turns out that many properties behave in a generic manner, i.e., the results can be phrased in terms independent of the order q of the field of definition. It is the aim of this article to present a framework in which this generic behaviour can be conveniently formulated (and partly also proved).

More precisely, assume we are given a family of groups of Lie type, like the groups GL_{n}(q) for fixed n but varying prime power q. Then a crucial role in the description of this family of groups is played by the Weyl group, which is the same for all members in the family, together with the action induced by the Frobenius morphism (the twisting). The Weyl group occurs in a natural way as a reflection group on the vector space generated by the coroots, and the Frobenius morphism induces an automorphism of finite order. This leads to the concept of a generic finite reductive group 𝔾. The orders of the groups attached to a generic finite reductive group 𝔾 can be obtained as values of one single polynomial, the order polynomial |𝔾| of 𝔾. In analogy to the concept of ℓ-subgroup of a finite group, for any cyclotomic polynomial Φ_{d} dividing the order polynomial |𝔾| one can define d-tori of 𝔾. These satisfy a complete analogue of the Sylow theorems for finite groups. These concepts were first formalized in [1].

In the representation theory of groups of Lie type, the right generic objects are the unipotent characters. It follows from results of Lusztig that they can be parametrized in terms only depending on the generic group 𝔾. Moreover, the functor of twisted induction can be shown to be generic. It turns out that for any d such that Φ_{d} divides |𝔾| there holds a d-Harish-Chandra theory for unipotent characters. This gives rise to a whole family of generalized Harish-Chandra theories which contains the usual Harish-Chandra theory for complex characters as the case d = 1. Surprisingly, the resulting generalized Harish-Chandra series allow a description of the subdivision into ℓ-blocks of the unipotent characters of a finite reductive group, at least for large primes ℓ, thus giving an application of the generic formalism to the ℓ-modular representation theory of groups of Lie type. The generalized Harish-Chandra series can be indexed by irreducible characters of suitable relative Weyl groups such that the decomposition of the functor of twisted induction can be compared to ordinary induction in these relative Weyl groups. This was proved in [3]. These relative Weyl groups turn out to be pseudo-reflection groups.

We introduce generic groups and their generic order in Sections 2 and 3 and formulate the d-Sylow theorems in Section 4. In Section 5 the setup and the main statements of ordinary Harish-Chandra theory are recalled. In Sections 6 and 7 we introduce generic unipotent characters and describe the generalized Harish-Chandra theories. The application to ℓ-blocks is given in Section 8. In the last section we discuss the decomposition of twisted induction and the role played in this by the relative Weyl group and (conjecturally) by its cyclotomic Hecke algebra.

2 Generic finite reductive groups

2.1 From algebraic groups to generic groups

We start with a connected reductive algebraic group G over the algebraic closure of a finite field of characteristic p > 0. We assume that G is already defined over a finite field and let F : G → G be the corresponding Frobenius morphism. The group of fixed points G^{F} is then a finite group of Lie type. Recall from Geck’s lecture that the choice of an F -stable maximal torus T of G and a Borel subgroup containing T gives rise to a root datum (X, R, Y, R^{∨}), consisting of the character and cocharacter groups X, Y of T, the set of roots R ⊂ X and the set of coroots R^{∨} ⊂ Y. The Frobenius map F acts on Y_{ℝ} := Y ⊗_{ℤ} ℝ as qϕ where q is a power of p and ϕ is an automorphism of finite order. Replacing the Borel subgroup by another one containing T changes ϕ by an element of the Weyl group W of G with respect to T. Hence ϕ is uniquely determined as automorphism of Y_{ℝ} up to elements of W.

We can thus naturally associate to (G, T, F) the data (X, R, Y, R^{∨}, W ϕ). Here,

(i) X, Y are free ℤ-modules of equal finite rank, endowed with a duality X × Y → ℤ, (x, y) ↦ ⟨x, y⟩,

(ii) R ⊂ X and R^{∨} ⊂ Y are root systems with a bijection R → R^{∨}, α ↦ α^{∨}, such that ⟨α, α^{∨}⟩ = 2 (see Section 6 in Rouquier’s lecture),

(iii) W is the Weyl group of the root system R^{∨} in Y and ϕ is an automorphism of Y of finite order stabilizing R^{∨}.

A quintuple (X, R, Y, R^{∨}, W ϕ) satisfying these properties is called a generic finite reductive group or complete root datum.

Conversely, let’s start from a complete root datum 𝔾 = (X, R, Y, R^{∨}, W ϕ). Then for any choice of a prime number p, 𝔾 determines a pair (G, T) as above, up to inner automorphisms of G induced by T (see the lecture of Carter). Moreover, the additional choice of a power q of p determines a triple (G, T, F) as above. In this way the generic finite reductive group 𝔾 gives rise to a whole series {𝔾(q) := G^{F} | q a prime power} of groups of Lie type.

Example 2.1 Let G = GL_{n}(\overline{𝔽_{q}}) be the group of invertible n×n-matrices over the algebraic closure of the finite field 𝔽_{q} with the maximal torus T consisting of the diagonal matrices in G. Then T is F -stable for the Frobenius map F : G → G which raises every matrix entry to its qth power, as well as for the product F^{−} of F with the transpose-inverse map on G. In the first case, the group of F -fixed points is the general linear group over 𝔽_{q}, while for the second Frobenius map F^{−} we obtain the general unitary group GU_{n}(q). One easily checks that (G, T, F) gives rise to a generic finite reductive group of the form

GL_{n} = (ℤ^{n}, R, ℤ^{n}, R^{∨}, 𝔖_{n} · Id)

with R = R^{∨} = {e_{i} − e_{j} | i ≠ j}, where {e_{1}, …, e_{n}} is the standard basis of ℤ^{n}, while (G, T, F^{−}) gives rise to GU_{n} = (ℤ^{n}, R, ℤ^{n}, R^{∨}, 𝔖_{n}(−Id)). In this sense we may think of GU_{n}(q) as being GL_{n}(−q).

2.2 Generic Levi subgroups and tori

A generic Levi subgroup of 𝔾 = (X, R, Y, R^{∨}, W_{𝔾}ϕ) is by definition a generic finite reductive group of the form

𝕃 = (X, R′, Y, R′^{∨}, W_{𝕃}wϕ) ,

where w ∈ W_{𝔾} and R′^{∨} is a wϕ-stable parabolic subsystem of R^{∨} with Weyl group W_{𝕃}. A generic torus is a generic finite reductive group with R = R^{∨} = ∅. Thus a generic torus of 𝔾 has the form (X′, Y′, (wϕ)|_{Y′}) for some w ∈ W_{𝔾} and some wϕ-stable direct summand Y′ of Y, where X′ is the dual of Y′, i.e., X′ = X/(Y′)^{⊥} with

(Y′)^{⊥} = {x ∈ X | ⟨x, y⟩ = 0 for all y ∈ Y′} .

The Weyl group W_{𝔾} acts naturally on the set of generic Levi subgroups of 𝔾 as well as on the set of generic tori of 𝔾. The following result allows to switch freely between the language of generic finite reductive groups and that of actual groups of Lie type (see [1, Th. 2.1]):

Proposition 2.2 Let 𝔾 be a generic finite reductive group and q a prime power, and let (G, T, F) be the associated finite reductive group.

(a) There exists a natural bijection between the W_{𝔾}-classes of generic Levi subgroups of 𝔾 and the G^{F} -classes of F -stable Levi subgroups of G.

(b) There exists a natural bijection between the W_{𝔾}-classes of generic tori of 𝔾 and the G^{F} -classes of F -stable tori of G

We sketch the construction of the bijection in (a). Let S ≤ G be an F - stable torus in G. Then S is contained in some F -stable maximal torus of G. Since all maximal tori of G are conjugate, there exists a g ∈ G with S ≤ ^{g}T. Then g^{−1}F (g) normalizes T, and we denote its image in the Weyl group W = N_{G}(T)/T by w. The cocharacter group Y′ of S^{g} can be identified with a subgroup of the cocharacter group Y of T, and similarly the character group X′ of S^{g} with a quotient of X. We now associate to S the generic torus 𝕊 := (X′, Y′, (wϕ)|_{Y′}) of 𝔾. Note that by construction Y′ is wϕ-stable. Conversely, if 𝕊 = (X′, Y′, (wϕ)|_{Y′}) is a torus of 𝔾, choose n ∈ N_{G}(T) with image w. The inclusion Y′ ≤ Y defines an nF -stable subtorus S_{w} of T. By the theorem of Lang (see Theorem 3.5 in Geck’s lecture) there exists a g ∈ G with n = g^{−1}F (g). Then S := ^{g}S_{w} is an F -stable torus of G.

Part (b) is essentially a consequence of the following connection between the concepts of generic tori and generic Levi subgroups. For a generic torus 𝕊 = (X′, Y′, (wϕ)|_{Y′}) of 𝔾 we define its centralizer to be the generic Levi subgroup

C_{𝔾}(𝕊) := (X, R′, Y, R′^{∨}, W′wϕ) ,

where R′ consists of those roots α ∈ R which are orthogonal to Y′, and W′ is the Weyl group of the root system R′^{∨}. For a generic Levi subgroup 𝕃 = (X, R′, Y, R′^{∨}, W_{𝕃}wϕ) we define its center to be the generic torus

Z(𝕃) := (X′, Y′, (wϕ)|_{Y′})

of 𝔾, where Y′ is the orthogonal of R′ in Y. With these definitions it is easy to see that any generic Levi subgroup 𝕃 of 𝔾 satisfies C_{𝔾}(Z(𝕃)) = 𝕃 (see [1, Prop. 1.3]). Furthermore, if S is an F -stable torus of the algebraic group G and L is its centralizer in G, then the generic group 𝕃 of L is the centralizer in 𝔾 in the sense defined before of the generic group 𝕊 of S, and similarly for the center of a Levi.

2.3 Some constructions

Many constructions for algebraic groups have a convenient formulation in terms of generic groups. Let 𝔾 = (X, R, Y, R^{∨}, W ϕ) be a generic group. The dual generic group (see Section 8 in the lecture of Geck) is defined by 𝔾^{∗} = (Y, R^{∨}, X, R, W ϕ^{∨}^{−1}), where ϕ^{∨} is the automorphism of X adjoint to ϕ. We write Q(R) for the ℤ-submodule of X generated by R. Then the radical of 𝔾 (see Section 2 in the lecture of Carter) is the generic torus

Rad(𝔾) := (X/Q(R)^{⊥⊥}, Q(R)^{⊥}, ϕ|_{Q(R)^{⊥}}) ,

the semisimple quotient of 𝔾 is

𝔾_{ss} := (Q(R)^{⊥⊥}, R, Y/Q(R)^{⊥}, R^{∨}, W ϕ) .

The reader is invited to verify in each case that our construction does indeed yield the generic group associated to the result of the corresponding operation on algebraic groups. Finally, for a positive integer d we denote by 𝔾^{(d)} the generic group (X^{d}, R^{d}, Y^{d}, (R^{∨})^{d}, W^{d}σϕ), where (X^{d}, R^{d}, Y^{d}, (R^{∨})^{d}) is the direct product of d copies of the root datum of 𝔾, and where σ is the automorphism of Y^{d} which permutes the d factors cyclically. One checks that 𝔾^{(d)}(q) ≅ 𝔾(q^{d}).

3 The polynomial order

We want to formulate a Sylow theory for generic groups. For this, we first have to say what should be the order |𝔾| of a generic group 𝔾, and then define what Φ-subgroups for prime divisors Φ of |𝔾| are.

3.1 The order polynomial

Let 𝔾 = (X, R, Y, R^{∨}, W ϕ) be a generic finite reductive group. The Weyl group W acts on V := Y ⊗_{ℤ} ℂ as a group generated by reflections. Thus its ring of invariants S(V)^{W} in the symmetric algebra S(V) is a polynomial ring by the theorem of Shephard/Todd and Chevalley (see Theorem 9.1 in Rouquier’s lecture). Since the element ϕ acts completely reducibly on V it is possible to choose n = dim(V) algebraically independent homogeneous invariants f_{1}, …, f_{n} ∈ S(V)^{W} generating S(V)^{W} which are eigenvectors for the action of ϕ. Thus ϕf_{i} = ζ_{i}f_{i} for some root of unity ζ_{i}. Denote the degree of f_{i} by d_{i}. Then it is known that the family {(d_{1}, ζ_{1}), …, (d_{n}, ζ_{n})} is independent of the choice of the f_{i} (see for example [12, 6.1]). Let

ϵ_{𝔾} := (−1)^{n}ζ_{1} … ζ_{n} = det(−ϕ) ∈ {±1}

and let x be an indeterminate. We define the polynomial order of 𝔾 to be

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

where N = |R|/2 is the number of reflections in W. For example, if 𝔾 is a torus, i.e., if R = R^{∨} = ∅, W = 1, then the above formula becomes |𝔾| = ϵ_{𝔾} det_{V} (1 − xϕ), hence the polynomial order of a torus (X, Y, ϕ) is just the characteristic polynomial of ϕ on V.

Proposition 3.1 We have |𝔾| = x^{N} ∏_{i=1}^{n}(x^{d_{i}} − ζ_{i}) ∈ ℤ[x].

In the case ϕ = 1 this is just Molien’s formula for the ring of invariants S(V)^{W}. In the general case it follows by evaluating the graded trace of elements wϕ on the graded W ⟨ϕ⟩-module S(V) in two ways.

It can be shown that for any choice of a prime power q the order of the corresponding finite group 𝔾(q) is given by

|𝔾(q)| = |𝔾|(q)

(see for example [5, Th. 9.4.10 and 14.3.1]). It is clear from Proposition 3.1 that |𝔾| is a product of cyclotomic polynomials, that is, there exist nonnegative integers a(d) such that

|𝔾|(x) = x^{N} ∏_{d} Φ_{d}(x)^{a(d)} ,

where Φ_{d}(x) denotes the dth cyclotomic polynomial over ℚ, i.e., the polynomial whose roots are the primitive dth roots of unity. These cyclotomic polynomials will be considered to be the primes dividing the order of 𝔾.

Example 3.2 We continue Example 2.1. For the generic group GL_{n} the order is given by

|GL_{n}| = ∏_{i=0}^{n−1}(x^{n} − x^{i}) = x^{n(n−1)/2} ∏_{d} Φ[\frac{n}{d}]_{d}

where [a] denotes the largest integer smaller than or equal to a. The formula for the polynomial order immediately shows that

|GU_{n}|(x) = ±|GL_{n}|(−x).

This is an instance of the so-called Ennola-duality between GL_{n} and GU_{n}.

3.2 d-tori and d-split Levi subgroups

A generic torus 𝕊 is called a d-torus if the order polynomial |𝕊|(x) is a power of Φ_{d}(x). Thus by our above remark 𝕊 = (X, Y, ϕ) is a d-torus if and only if all eigenvalues of ϕ on V are primitive dth roots of unity. Translated to algebraic groups, a torus 𝕊 is d-split if and only if the corresponding finite reductive groups 𝕊(q) are the groups of F -fixed points of an algebraic torus S which splits over 𝔽_{q^{d}} but no subtorus of which splits over any proper subfield.

The centralizers of generic d-tori of 𝔾 are called d-split Levi subgroups. Note that, in particular, 𝔾 itself is d-split. Thus any d such that Φ_{d} divides |𝔾| gives rise to two families of substructures of 𝔾: d-tori and d-split Levi subgroups.

Example 3.3 (a) Let 𝔾 be a generic finite reductive group and (G, F) a finite reductive group determined from 𝔾 and a choice of a prime power q. Then the 1-split Levis of 𝔾 are precisely those generic Levi subgroups 𝕃 of 𝔾 such that the corresponding F -stable Levi subgroup L of G is contained in an F -stable parabolic subgroup of G. Thus the case d = 1 describes what is usually called the split or Harish-Chandra Levi subgroups. Similarly, the 1-tori are those generic tori 𝕊 such that the corresponding F -stable torus S of G is split in the usual sense.

(b) We continue Example 2.1. If (X′, Y′, w|_{Y′}) is a d-torus of GL_{n} for d = n then at least one eigenvalue of w on V has to be a primitive nth root of unity. The only elements with this property of W = 𝔖_{n} in its permutation representation are the n-cycles (i.e., the Coxeter elements). Thus any nontrivial n-torus of GL_{n} is conjugate to 𝕊 = (X′, Y′, w|_{Y′}) where w is an n-cycle and Y′ is the maximal sublattice of Y = ℤ^{n} such that all eigenvalues of w on Y′ ⊗_{ℤ} ℂ are primitive n-th roots of unity (so Y′ has rank φ(n)). The centralizer of 𝕊 is the Coxeter torus (X, Y, w), which according to our above definition is just the generic group GL_{1}^{(n)}. More generally, it is easily seen by considering cycle shapes of elements in 𝔖_{n} that for arbitrary d the d-split Levi subgroups of GL_{n} have the type

GL_{n_{1}}^{(d)} × … × GL_{n_{r}}^{(d)} × GL_{s}

for some n_{1}, …, n_{r}, s with d(n_{1} + … + n_{r}) + s = n.

The finite reductive groups associated to d-tori have a very simple structure (see [1, Prop. 3.3]):

Proposition 3.4 Let 𝕊 be a generic d-torus of order Φ_{d}^{a(d)}. Let 𝕊(q) be the finite reductive group associated to 𝕊 by the choice of a prime power q. Then 𝕊(q) is the direct product of a(d) cyclic groups of order Φ_{d}(q).

4 d-Sylow theorems

The d-tori introduced in the previous section behave like ℓ-subgroups of finite groups. To make this more precise we need one more definition. For a Levi subgroup 𝕃 of a generic group 𝔾 we define the relative Weyl group of 𝕃 in 𝔾 as

W_{𝔾}(𝕃) := N_{W_{𝔾}}(W_{𝕃})/W_{𝕃} .

Theorem 4.1 Let 𝔾 be a generic finite reductive group and d such that Φ_{d}(x) divides |𝔾|.

(a) There exist non-trivial d-tori of 𝔾.

(b) For any maximal d-torus 𝕊 of 𝔾 we have |𝕊| = Φ_{d}^{a(d)}, where a(d) is the precise power of Φ_{d} dividing |𝔾|.

(c) Any two maximal d-tori of 𝔾 are conjugate.

(d) Let 𝕊 be a maximal d-torus of 𝔾 and 𝕃 = C_{𝔾}(𝕊) its centralizer. Then

\frac{|𝔾|}{|W_{𝔾}(𝕃)||𝕃|} ≡ 1 (mod Φ_{d}) in ℤ[x].

Replacing ‘generic finite reductive group’ by ‘finite group’, ‘d-torus’ by ‘ℓsubgroup’ and Φ_{d} by ℓ the first three statements of the theorem just become the familiar Sylow theorems for finite groups. In view of this, the maximal d-tori of 𝔾 are called generic Sylow d-tori. The fourth assertion is the translation of the fact that the number of Sylow ℓ-subgroups of a finite group is congruent to 1 modulo ℓ.

We sketch the proof of the first three parts of this theorem. Since the order of a torus (X, Y, ϕ) is just the characteristic polynomial of ϕ on V = Y ⊗_{ℤ} ℂ we may translate assertions (a) to (c) to assertions about elements of W_{𝔾} as follows: for w ∈ W_{𝔾} and ζ ∈ ℂ let

V (wϕ, ζ) := {v ∈ V | wϕ v = ζv}

be the eigenspace of wϕ on V for the eigenvalue ζ. Then (a) states that for any d with a(d) ≠ 0 there exist w ∈ W_{𝔾} with V (wϕ, ζ_{d}) ≠ 0 where ζ_{d} is a primitive dth root of unity. The second part claims that there exist w such that dim(V (wϕ, ζ_{d})) = a(d), and the third part claims that all these maximal eigenspaces for a fixed ζ_{d} are conjugate under W_{𝔾}. It was shown by Springer [12, 3.4 and 6.2] that these three statements do hold for arbitrary finite groups generated by reflections, hence in particular for Weyl groups. The proof of the last part is of a somewhat different flavour (see [1, Th. 3.4]) and we will not go into this here.

Example 4.2 Let 𝔾 = (X, R, Y, R^{∨}, W ϕ) be the generic finite reductive group attached to the series of Steinberg triality groups ^{3}D_{4}(q). Thus W is the Weyl group of type D_{4} and ϕ is a non-trivial automorphism of W of order 3 induced by the inclusion of W into the Weyl group of type F_{4}. The Weyl group of type D_{4} has polynomial invariants generated by homogeneous elements of degrees 2, 4, 6, 4. The corresponding eigenvalues of ϕ are given by 1, ζ_{3}, 1, ζ_{3}^{2}, with ζ_{3} a primitive third root of unity. Thus 𝔾 has polynomial order

|𝔾| = x^{12}(x^{2} − 1)(x^{8} + x^{4} + 1)(x^{6} − 1) = x^{12}Φ_{1}^{2}Φ_{2}^{2}Φ_{3}^{2}Φ_{6}^{2}Φ_{12} .

According to the Sylow theorems there exist (maximal) tori of 𝔾 of orders Φ_{3}^{2}, Φ_{6}^{2}, Φ_{12}, namely the Sylow d-tori for d = 3, 6, 12. Furthermore, there exist Sylow tori of orders Φ_{1}^{2}, Φ_{2}^{2}, but these tori are not maximal.

Let 𝔾 be a generic group and q a prime power. The Sylow Theorem 4.1 also translates to an assertion about Sylow ℓ-subgroups of 𝔾(q) for large primes ℓ not dividing q (see [1, Cor. 3.13]):

Proposition 4.3 Let ℓ be a prime dividing |𝔾(q)| but not dividing q|W ⟨ϕ⟩|. (a) There exists a unique d such that ℓ|Φ_{d}(q) and Φ_{d}||𝔾|.

(b) Any Sylow ℓ-subgroup of 𝔾(q) is contained in the group 𝕊(q) for some Sylow d-torus 𝕊 of 𝔾.

(c) The Sylow ℓ-subgroups are isomorphic to a direct product of a(d) cyclic groups of order ℓ^{a}, where Φ_{d}^{a(d)} is the precise power of Φ_{d} dividing |𝔾| and where ℓ^{a} is the precise power of ℓ dividing Φ_{d}(q).

The first part is an easy exercise, while the second and third follow from Proposition 3.4 and Theorem 4.1.

5 Ordinary Harish-Chandra theory

We leave the generic point of view for a moment in order to give a brief account of Harish-Chandra theory for complex characters of finite groups of Lie type. In the next section we will return to the generic setting and explain how this ordinary Harish-Chandra theory can be considered as a special case of a whole family of such theories.

Let G be a connected reductive algebraic group defined over a finite field and let F : G → G be the corresponding Frobenius morphism. Let P be an F -stable parabolic subgroup of G, with F -stable Levi-decomposition P = UL, i.e., U is the unipotent radical of P and L is an F -stable Levi complement. Any ℂL^{F} -module M can be considered as a ℂP^{F} -module via the projection map P^{F} → L^{F}. The induced module of M from P^{F} to G^{F} is denoted by R_{L,P}^{G}(M) (see Remark 7.2 in Geck’s lecture). Conversely, for a ℂG^{F} -module M we denote by ^{∗}R_{L,P}^{G}(M) the ℂL^{F} -module of U^{F} -fixed points of M. It was first shown by Deligne that for fixed L the result of both constructions is independent of the choice of parabolic subgroup P containing L. Let L be the set of F -stable Levi subgroups of F -stable parabolic subgroups of G. Thus for any L ∈ L we have linear maps

R_{L}^{G} : ℤIrr(L^{F}) → ℤIrr(G^{F}), ^{∗}R_{L}^{G} : ℤIrr(G^{F}) → ℤIrr(L^{F}),

called Harish-Chandra induction and restriction. These are adjoint to each other with respect to the scalar product of characters. Moreover, Harish-Chandra induction is transitive in the sense that R_{M}^{G}∘R_{L}^{M} = R_{L}^{G} for L, M ∈ L with L ⊂ M (see for example [6, Prop. 4.7]) and it satisfies a Mackey-formula (see [6, Th. 5.1]).

A key notion in Harish-Chandra theory is that of cuspidality. An irreducible character χ ∈ Irr(G^{F}) is called cuspidal if ^{∗}R_{L}^{G}(χ) = 0 for all proper Levi subgroups L ∈ L. We then also say that the pair (G, χ) is cuspidal. Let

K := {(L, λ) | L ∈ L, λ ∈ Irr(L^{F})} .

For (L_{1}, λ_{1}), (L_{2}, λ_{2}) ∈ K we write (L_{1}, λ_{1}) ≤ (L_{2}, λ_{2}) if L_{1} ⊂ L_{2} and λ_{2} is a constituent of R_{L_{1}}^{L_{2}}(λ_{1}). This defines a partial order relation on the set of pairs K. By the adjointness of R_{L}^{G} and ^{∗}R_{L}^{G} the minimal elements are just the cuspidal pairs. The following Harish-Chandra principle is proved for example in [6, Sec. 6]:

Theorem 5.1 (a) (Disjointness) For each γ ∈ Irr(G^{F}) there exists a unique cuspidal pair (L, λ) (up to G^{F} -conjugacy) such that γ occurs in R_{L}^{G}(λ).

(b) (Transitivity) Let (L, λ) be cuspidal and (L, λ) ≤ (M, μ), (M, μ) ≤ (G, γ). Then (L, λ) ≤ (G, γ).

The first part is a consequence of the Mackey formula for Harish-Chandra induction and restriction, while part (b) follows immediately from the transitivity of R_{L}^{G} stated above and the fact that it sends characters to characters.

For a cuspidal pair (L, λ) ∈ K we define its Harish-Chandra series to be

Irr(G^{F} , (L, λ)) := {γ ∈ Irr(G^{F}) | γ occurs in R_{L}^{G}(λ)}.

Hence Theorem 5.1(a) provides a partition

Irr(G^{F}) = ∐_{(L,λ)} Irr(G^{F} , (L, λ))

of Irr(G^{F}) into disjoint Harish-Chandra series where the union runs over cuspidal pairs in K up to G^{F} -conjugation.

Much more can be said about ordinary Harish-Chandra series. For cuspidal pairs (L, λ) ∈ K the relative Weyl group

W_{G^{F}} (L, λ) := N_{G^{F}} (L, λ)/L^{F}

is a finite Coxeter group in a natural way. Let M be a ℂL^{F} -module affording the character λ. Howlett/Lehrer [8] have shown that the endomorphism algebra of R_{L}^{G}(M) is an Iwahori-Hecke algebra (see Section 8 in Rouquier’s lecture) of the Coxeter group W_{G^{F}} (L, λ), possibly twisted by a 2-cocycle. Thus there is a natural bijection

I_{(L,λ)}^{G^{F}} : Irr(W_{G^{F}} (L, λ)) → Irr(G^{F} , (L, λ))

between the set of complex irreducible characters of W_{G^{F}} (L, λ) and the Harish-Chandra series Irr(G^{F} , (L, λ)). Moreover, the decomposition of R_{L}^{G}(λ) agrees with the image under I_{(L,λ)}^{G^{F}} of the decomposition of the regular character of the relative Weyl group W_{G^{F}} (L, λ). This statement will be made more precise in Theorem 9.1 in the general context of d-Harish-Chandra theories.

6 Generic unipotent characters

In the previous sections we have introduced the concept of generic groups to capture much of the generic nature of the subgroup structure of finite reductive groups. We now turn to the description of the generic behaviour of irreducible characters. The results in this second part are considerably deeper than those presented in the first part, hence we will not be able to say very much about the proofs.

Recall from the lecture of Geck that an irreducible character γ ∈ Irr(G^{F}) is called unipotent if there exists an F -stable maximal torus S ≤ G such that γ is a constituent of R_{S}^{G}(1). Let U(G^{F}) denote the set of unipotent characters of G^{F}. The results of Lusztig on unipotent characters sketched in Sections 9 and 10 of Geck’s lecture may be rephrased as follows: there exists a set U(𝔾) and a map

Deg : U(𝔾) → ℚ[x], γ ↦ Deg(γ),

such that for any choice of p and q (and hence of G and F) there is a bijection ψ_{q}^{𝔾} : U(𝔾) → U(G^{F}) such that ψ_{q}^{𝔾}(γ) has degree ψ_{q}^{𝔾}(γ)(1) = Deg(γ)(q). The set U(𝔾) is called the set of (generic) unipotent characters of 𝔾.

Also recall from the lecture of Geck that for any F -stable Levi subgroup L ≤ G Deligne and Lusztig defined functors

R_{L}^{G} : ℤIrr(L^{F}) → ℤIrr(G^{F}), ^{∗}R_{L}^{G} : ℤIrr(G^{F}) → ℤIrr(L^{F}),

between the character groups of L^{F} and G^{F}, adjoint to each other with respect to the usual scalar product of characters. (More precisely, the definition of these functors also involves the choice of a parabolic subgroup P containing L, but in our situation it turns out that they are in fact independent of this choice.) It follows from results of Shoji (see [3, Ths. 1.26 and 1.33]) that the maps R_{L}^{G} and ^{∗}R_{L}^{G} are generic:

Theorem 6.1 For any generic Levi subgroup 𝕃 of 𝔾 there exist linear maps

R_{𝕃}^{𝔾} : ℤU(𝕃) → ℤU(𝔾), ^{∗}R_{𝕃}^{𝔾} : ℤU(𝔾) → ℤU(𝕃),

satisfying ψ_{q}^{𝔾}∘R_{𝕃}^{𝔾} = R_{L}^{G}∘ψ_{q}^{𝕃} for all q (when extending ψ_{q}^{𝔾} linearly to ℤU(𝔾)).

In this sense, the sets of unipotent characters together with the collection of functors R_{L}^{G} and ^{∗}R_{L}^{G} are generic for a series of groups of Lie type.

The Jordan decomposition of irreducible characters of G^{F} (see Section 9 in Geck’s lecture) shows the prominent role of unipotent characters in the description of Irr(G^{F}). We will hence restrict our attention to the subset U(G^{F}) ⊂ Irr(G^{F}), respectively to its generic version U(𝔾).

Example 6.2 In the case of the finite reductive group G^{F} = GL_{n}(q) the unipotent characters are just the constituents of the permutation character 1_{B^{F}}^{G^{F}} on the F -fixed points of an F -stable Borel subgroup B of G. The endomorphism algebra of this permutation module is the Hecke algebra of type A_{n−1} with parameter q (see Section 8 in the lecture of Rouquier). Its irreducible characters are in bijection with the irreducible characters of the symmetric group 𝔖_{n}, the Weyl group of G^{F}. Since the latter are naturally indexed by partitions α ⊢ n, the same is true for the unipotent characters of G^{F}. Hence in this case we have U(GL_{n}) = {γ_{α} | α ⊢ n}. The function Deg : U(GL_{n}) → ℚ[x] can be described as follows: for a partition α = (α_{1} ≤ … ≤ α_{m}) of n let β_{i} := α_{i} + i − 1, 1 ≤ i ≤ m. Then

Deg(γ_{α}) = \frac{(x − 1) … (x^{n} − 1) ∏_{j>i}(x^{β_{j}} − x^{β_{i}})}{x^{\binom{m−1}{2}+\binom{m−2}{2}+...} ∏_{i} ∏_{j=1}^{β_{i}}(x^{j} − 1)} .

(Despite appearance, this always is a polynomial.) Furthermore, there exists a bijection

U(GL_{n}) → U(GU_{n}), γ ↦ γ^{−},

such that Deg(γ^{−})(x) = ±Deg(γ)(−x). This is another consequence of the Ennola duality between GL_{n} and GU_{n}.

7 d-Harish-Chandra theories

The idea that a generalized Harish-Chandra theory should exist for the unipotent characters of a finite group of Lie type first occurred in the papers by Fong/Srinivasan [7] and Schewe [11] in particular cases. The general case was settled in [3].

7.1 d-cuspidal characters

Let 𝔾 be a generic finite reductive group and d an integer such that a(d) > 0, i.e., such that Φ_{d} divides |𝔾|. We first have to introduce a generalization of cuspidal characters:

Definition 7.1 A generic unipotent character γ ∈ U(𝔾) is called d-cuspidal if ^{∗}R_{𝕃}^{𝔾}(γ) = 0 for all d-split Levi subgroups 𝕃 properly contained in 𝔾.

Thus in the case d = 1 the 1-cuspidal characters of 𝔾 are those whose image under ψ_{q}^{𝔾} is a cuspidal unipotent character as defined in Section 5. We have the following alternative characterization of d-cuspidality (see [3, Prop. 2.9]):

Proposition 7.2 A unipotent character γ ∈ U(𝔾) is d-cuspidal if and only if Deg(γ)_{d} = |𝔾_{ss}|_{d}.

Here, we write f_{d} for the Φ_{d}-part of f ∈ ℚ[x]. The semisimple quotient 𝔾_{ss} of 𝔾 was defined in Section 2.

Example 7.3 We continue Example 6.2. Let α = (α_{1} ≤ … ≤ α_{m}) be a partition of n. A d-hook of α is a pair h = (ν, β) of integers 0 ≤ ν < β such that β occurs among the {α_{i} + i − 1 | 1 ≤ i ≤ m} but ν does not. The length of the hook h is l(h) = β − ν and h is also called an l(h)-hook. A moments thought shows that up to a power of x the degree given in Example 6.2 of the unipotent character γ_{α} of the generic group of type GL_{n} is of the form

\frac{(x − 1) … (x^{n} − 1)}{∏_{h hook of α}(x^{l(h)} − 1)} .

Note that the existence of an ad-hook of α for some a ≥ 1 implies the existence of a d-hook. Thus, by Proposition 7.2 the unipotent character γ_{α} is d-cuspidal if and only if α has no d-hook. Such partitions α are also called d-cores. In particular, GL_{n} has a 1-cuspidal unipotent character if and only if n = 0, since the empty partition is the only 1-core. This shows that indeed all unipotent characters of GL_{n} occur in R_{𝕊}^{𝔾}(1) for the maximally split torus 𝕊.

7.2 d-Harish-Chandra series

A pair (𝕃, λ) consisting of a d-split Levi subgroup 𝕃 of 𝔾 and a unipotent character λ ∈ U(𝕃) is called d-split. It is called d-cuspidal if moreover λ is d-cuspidal. We introduce the following relation on the set of d-split pairs:

Definition 7.4 Let (𝕄_{1}, μ_{1}) and (𝕄_{2}, μ_{2}) be d-split in 𝔾. Then we say that (𝕄_{1}, μ_{1}) ≤_{d} (𝕄_{2}, μ_{2}) if 𝕄_{1} is a d-split Levi subgroup of 𝕄_{2} and μ_{2} occurs in R_{𝕄_{1}}^{𝕄_{2}}(μ_{1}).

For a d-cuspidal pair (𝕃, λ) of 𝔾 we write

U(𝔾, (𝕃, λ)) := {γ ∈ U(𝔾) | (𝕃, λ) ≤_{d} (𝔾, γ)}

for the set of unipotent characters of 𝔾 lying above (𝕃, λ). We call U(𝔾, (𝕃, λ)) the d-Harish-Chandra series above (𝕃, λ) because of the following fundamental result, which shows that for any d we obtain a generalized Harish-Chandra theory (in generalization of the case d = 1 in Theorem 5.1):

Theorem 7.5 Let 𝔾 be a generic finite reductive group and d ≥ 1 such that Φ_{d} divides |𝔾|.

(a) (Disjointness) The sets U(𝔾, (𝕃, λ)) (where (𝕃, λ) runs over a system of representatives of the W_{𝔾}-conjugacy classes of d-cuspidal pairs) form a partition of U(𝔾).

(b) (Transitivity) Let (𝕃, λ) be d-cuspidal and (𝕄, μ) be d-split such that (𝕃, λ) ≤_{d} (𝕄, μ) and (𝕄, μ) ≤_{d} (𝔾, γ). Then (𝕃, λ) ≤_{d} (𝔾, γ).

The statement of Theorem 7.5 has been checked case by case (except for the case d = 1 where a conceptual proof is known). It is a consequence of the more precise Theorem 9.1 about the decomposition of twisted induction of unipotent characters from d-split Levi subgroups. Note that, in contrast to the situation for Theorem 5.1, part (b) is far from obvious since R_{L}^{G} does not necessarily take characters to actual characters.

Example 7.6 We continue Example 7.3, where we saw that the d-cuspidal characters of GL_{n} are indexed by partitions of n which are d-cores. Let h = (ν, β) be a d-hook of the partition α = (α_{1} ≤ … ≤ α_{m}), and let j be the index with α_{j} + j − 1 = β. Then α′ = (α′_{1} ≤ … ≤ α′_{m}) is called the partition obtained from α by removing the d-hook h if the set {α′_{i} + i − 1} coincides with the set

{α_{i} + i − 1 | i ≠ j} ∪ {α_{j} + j − 1 − (β − ν)}.

The d-core obtained from α by successively removing all possible d-hooks is called the d-core of α. (The reader may want to check that the d-core does not depend on the order in which d-hooks are removed and is hence welldefined.) It can be shown that the d-Harish-Chandra series of GL_{n} above the d-cuspidal character indexed by the d-core α consists of the unipotent characters indexed by partitions of n whose d-core is α.

In Example 3.3(b) we determined the structure of d-split Levi subgroups of GL_{n}. By Example 7.3 the generic group GL_{n_{i}}^{(d)} has a d-cuspidal unipotent character only if n_{i} = 1. Thus the d-cuspidal pairs of GL_{n} are precisely the pairs (𝕃, λ_{α}) where

𝕃 = GL_{1}^{(d)} × … × GL_{1}^{(d)} × GL_{s}

(r factors GL_{1}^{(d)}) with n = dr + s, and λ_{α} is the unipotent character of 𝕃 parametrized by the d-core α ⊢ s.

8 Generic blocks

One importance of d-Harish-Chandra series lies in their connection with ℓblocks of finite groups of Lie type for primes ℓ not dividing q. Before continuing the exposition of the generic theory we therefore briefly explain this application.

We fix 𝔾 and a choice of a prime power q, hence a pair (G, F). A prime ℓ not dividing q is called large for 𝔾 if ℓ does not divide the order of W ⟨ϕ⟩. If ℓ is large then by Proposition 4.3(a) there exists a unique d such that ℓ|Φ_{d}(q) and Φ_{d}||𝔾|.

Theorem 8.1 (Broué/Malle/Michel) Let ℓ be large for 𝔾, and assume that ℓ|Φ_{d}(q), Φ_{d}||𝔾|. Then the partition of U(G^{F}) into ℓ-blocks coincides with the image under ψ_{q}^{𝔾} of the partition of U(𝔾) into d-Harish-Chandra series. In particular, the distribution of unipotent characters of G^{F} into ℓ-blocks is generic.

This is proved in [3, Th. 5.24]. Let us write b_{ℓ}(L, λ) for the unipotent ℓblock of G^{F} indexed by (𝕃, λ). The unipotent blocks and their defect groups can be described more precisely. For this we need:

Definition 8.2 For γ ∈ U(𝔾) let S_{d}(γ) denote the set of d-tori of 𝔾 contained in a maximal torus 𝕊 of 𝔾 such that ^{∗}R_{𝕊}^{𝔾}(γ) ≠ 0. The maximal elements of S_{d}(γ) are called the d-defect tori of γ.

The d-defect tori of a unipotent character can be characterized as follows (see [3, Th. 4.8]):

Proposition 8.3 Let (𝕃, λ) be a d-cuspidal pair and γ ∈ U(𝔾, (𝕃, λ)). Then the d-defect tori of γ are conjugate to Rad(𝕃)_{d}.

Here Rad(𝕃)_{d} denotes the Sylow d-torus of the torus Rad(𝕃). For an F -stable Levi subgroup L of G denote by Ab_{ℓ}Irr(L^{F}) the group of characters (over a splitting field of characteristic 0) of ℓ-power order of the abelian group L^{F} /[L^{F} , L^{F}]. Then [3, Th. 5.24] gives the following sharpening of Theorem 8.1:

Theorem 8.4 (Broué/Malle/Michel) Let ℓ be large for 𝔾, and assume that ℓ|Φ_{d}(q), Φ_{d}||𝔾|. Let (𝕃, λ) be a d-cuspidal pair of 𝔾.

(a) The ℓ-block b_{ℓ}(L, λ) of G^{F} consists of the irreducible constituents of the virtual characters R_{L}^{G}(θλ), where θ ∈ Ab_{ℓ}Irr(L^{F}).

(b) The defect groups of b_{ℓ}(L, λ) are the Sylow ℓ-subgroups of the groups of F -fixed points of the d-defect tori of unipotent characters in b_{ℓ}(L, λ).

The structure of the Sylow ℓ-subgroup of the group of F -fixed points of a d-torus was described in Proposition 4.3(c).

Example 8.5 Theorem 8.4 and the description of d-Harish-Chandra series for GL_{n} in Example 7.6 provide the following description of unipotent ℓ-blocks of GL_{n}(q) for large primes ℓ dividing Φ_{d}(q). Two unipotent characters γ_{α}, γ_{α′} lie in the same ℓ-block if and only if the d-cores of α and α′ coincide. (This was first proved by Fong and Srinivasan.) This result is very much reminiscent of the description of ℓ-blocks in the symmetric group 𝔖_{n}.

9 Relative Weyl groups

The d-Harish-Chandra theories presented in Section 7 seem to be just the shadow of a much deeper theory describing the decomposition of the functor of twisted induction. Much of this is still conjectural.

9.1 The decomposition of R_{𝕃}^{𝔾}

The only known proof of Theorem 7.5 in the case d ≠ 1 consists in determining the decomposition of the Deligne-Lusztig induced of d-cuspidal unipotent characters. To state this result we need to introduce an important invariant of a d-Harish-Chandra series. Let (𝕃, λ) be a d-cuspidal pair in 𝔾. Let (G, F) be a finite reductive group associated to 𝔾, and let L be an F -stable Levi subgroup of G corresponding to 𝕃 according to Proposition 2.2(b). Then the relative Weyl group W_{G^{F}} (L) = N_{G^{F}} (L)/L^{F} acts on Irr(L^{F}). By results of Lusztig, this leaves the subset U(L^{F}) of unipotent characters invariant. Moreover, the action on U(L^{F}) is generic in the sense that it is possible to define an action of W_{𝔾}(𝕃) = N_{W_{𝔾}}(W_{𝕃})/W_{𝕃} on U(𝕃) which, under all ψ_{q}^{𝔾}, specializes to the action of W_{G^{F}} (L) on U(L^{F}) (see [3]). This gives sense to the definition

W_{𝔾}(𝕃, λ) := N_{W_{𝔾}}(W_{𝕃}, λ)/W_{𝕃}

of the relative Weyl group of (𝕃, λ) in 𝔾.

Theorem 9.1 (Broué/Malle/Michel) For each d there exists a collection of isometries

I_{(𝕃,λ)}^{𝕄} : ℤIrr(W_{𝕄}(𝕃, λ)) → ℤU(𝕄, (𝕃, λ)),

such that for all 𝕄 and all (𝕃, λ) we have

R_{𝕄}^{𝔾} ∘ I_{(𝕃,λ)}^{𝕄} = I_{(𝕃,λ)}^{𝔾} ∘ Ind_{W_{𝕄}(𝕃,λ)}^{W_{𝔾}(𝕃,λ)} .

Here 𝕄 runs over the d-split Levi subgroups of 𝔾 and (𝕃, λ) over the set of d-cuspidal pairs of 𝕄.

This was proved in [3, Th. 3.2] by using results of Asai on the decomposition of R_{L}^{G} in the case of classical groups, and by explicit determination of these decompositions in the case of exceptional groups.

An isometry I from ℤIrr(W_{𝔾}(𝕃, λ)) to ℤU(𝔾, (𝕃, λ)) is nothing else but a bijection

Irr(W_{𝔾}(𝕃, λ)) ⟶^{∼} U(𝔾, (𝕃, λ)), χ ↦ γ_{χ},

together with a collection of signs

{ϵ(γ) | γ ∈ U(𝔾, (𝕃, λ))},

such that

I(χ) = ϵ(γ_{χ}) γ_{χ} .

Thus, Theorem 9.1 states that up to an adjustment by suitable signs, twisted induction from d-split Levi subgroups is nothing but ordinary induction in relative Weyl groups. As a consequence of [3, Th. 5.24] we moreover have the following congruence of character degrees

ϵ(γ_{χ}) Deg(γ_{χ}) ≡ χ(1) (mod Φ_{d}) in ℚ[x]

in the situation of Theorem 9.1.

Example 9.2 We continue the Example 4.2 of generic groups of type ^{3}D_{4}. The relative Weyl groups W_{𝔾}(𝕊_{d}) for the Sylow d-tori 𝕊_{d} with d ∈ {3, 6} turn out to be isomorphic to SL_{2}(3). Since SL_{2}(3) has 7 irreducible characters Theorem 9.1 implies that U(𝔾, (𝕊_{d}, 1)) has cardinality 7 for d ∈ {3, 6}. Moreover, the vector of degrees of the irreducible characters of SL_{2}(3) is (1, 1, 1, 2, 2, 2, 3), so R_{𝕊_{d}}^{𝔾}(1) contains three constituents with multiplicity ±1, three with multiplicity ±2 and one with multiplicity ±3.

9.2 Relative Weyl groups are pseudo-reflection groups

Theorem 9.1 turns attention towards the relative Weyl groups of d-cuspidal pairs (𝕃, λ). We stated in Section 5 that in the case d = 1 the relative Weyl groups turn out to be Coxeter groups, that is, groups with a faithful representation as a group generated by real reflections. In a case-by-case analysis the following surprising fact can be verified (see [2]):

Proposition 9.3 Let 𝔾 be a generic group and d a positive integer. For any d-cuspidal pair (𝕃, λ) of 𝔾 the relative Weyl group W_{𝔾}(𝕃, λ) is a pseudoreflection group, i.e., a finite group having a faithful representation as a group generated by complex reflections.

(See Section 9 of Rouquier’s talk for an introduction to pseudo-reflection groups.) In the case that 𝕃 is a maximal torus (and hence λ = 1) this is a result of Springer [12], but no a-priori proof is known in the general case.

Remark 9.4 It even turns out that, if W acts irreducibly on V = Y ⊗_{ℤ}ℂ, then the pseudo-reflection groups W_{𝔾}(𝕃, λ) are also irreducible in their natural reflection representation.

Example 9.5 Let 𝔾 be a generic group of type E_{7} and let d = 4. A Sylow 4-torus 𝕊 of 𝔾 has order Φ_{4}^{2} and its centralizer is a 4-split Levi subgroup 𝕃 = C_{𝔾}(𝕊) with semisimple part (PGL_{2})^{3}. Since 𝕃 is minimal 4-split all its unipotent characters are 4-cuspidal. The relative Weyl group W_{𝔾}(𝕃) is a twodimensional complex reflection group of order 96, denoted G_{8} by Shephard and Todd. It has two orbits of length 3 and two fixed points on the set U(𝕃). In particular, the relative Weyl group W_{𝔾}(𝕃, λ) for a (4-cuspidal) character λ in one of the orbits of length 3 is strictly smaller than W_{𝔾}(𝕃). It turns out to be the imprimitive complex reflection group denoted G(4, 1, 2) of order 32.

The relative Weyl groups occurring in generic finite reductive groups of exceptional type are collected in [3, Tables 1 and 3] while those for classical types are described in [3, Sec. 3], see also [2, 3B].

9.3 Cyclotomic Hecke algebras

We saw in Section 5 that in the ordinary Harish-Chandra case d = 1, the assertion of Theorem 9.1 has the following explanation. The endomorphism algebra of a Harish-Chandra induced cuspidal module with character λ ∈ Irr(L^{F}) is a Hecke algebra, a deformation of the group algebra of the relative Weyl group W_{G^{F}} (L, λ). This yields an indexation of the Harish-Chandra series Irr(G^{F} , (L, λ)) by Irr(W_{G^{F}} (L, λ)). It is tempting to conjecture that for arbitrary d a similar deep reason lies behind the result of Theorem 9.1. A conjecture in this direction was put forward in [2]. The only situation where this general conjecture is known to be true is that where 𝕃 is a Coxeter torus of 𝔾, by work of Lusztig [9]. Apart from that case, at present only some consequences of this deep conjecture have been verified.

More precisely, in the paper [2] we introduced certain deformations of group algebras of complex reflection groups, called cyclotomic Hecke algebras. Given an irreducible complex reflection group W, there are two possible definitions for the cyclotomic algebra H(W). The first is by generators and relations obtained by deforming a Coxeter like presentation of W. This approach, which is analogous to the usual definition of the Iwahori-Hecke algebra attached to a finite Coxeter group, was taken in [2]. The alternative is the construction of H(W) as a natural quotient of the group algebra of the topological braid group B_{W} associated to W, i.e., of the fundamental group of the space of regular orbits of W in its reflection representation (see Section 10 in Rouquier’s lecture). The equivalence of the two definitions was shown by Deligne and Brieskorn in the case that G is a real reflection group, and by Broué/Malle/Rouquier [4] in almost all complex cases.

Conjecturally, these cyclotomic algebras should be the correct replacement for Iwahori-Hecke algebras in the explanation of the result of Theorem 9.1 for the case d = 1. They form the object of some current research, see for example [4] and [10] and the references cited there. Unfortunately, we cannot go into more details about this fascinating subject.

References

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