CENTERS AND SIMPLE MODULES FOR IWAHORI-HECKE ALGEBRAS

MEINOLF GECK & RAPHAËL ROUQUIER

1. Introduction

The work of Dipper and James on Iwahori-Hecke algebras associated with the finite Weyl groups of type A_{n} has shown that these algebras behave in many ways like group algebras of finite groups. Moreover, there are “generic” features in the modular representation theory of these algebras which, at present, can only be verified in examples by explicit computations. This paper arose from an attempt to provide a conceptual explanation of these phenomena, in the general framework of the representation theory of (symmetric) algebras. We will study relations between the center of such algebras and properties of decomposition maps, and we will use this to obtain a general result about the “genericity” of the number of simple modules of Iwahori-Hecke algebras.

Usually, the formalism of decomposition maps is developed for algebras over a complete discrete valuation ring. However, in our applications to Iwahori-Hecke algebras, we have to make sure that this also works over the ring of Laurent polynomials in one indeterminate over the integers. Roughly speaking, this will be achieved by using the theory of Henselian rings (see [Ray]). In Section 2, we describe such a general setting for decomposition maps of algebras over integrally closed ground rings (see Proposition 2.11). Furthermore, we extend the standard results on the “Brauer-Cartan triangle” to the case of orders in non-semi-simple and non-split algebras, by using enlargements of the usual Grothendieck groups. As a formal consequence of the definition, we get a factorization property of decomposition maps (see Proposition 2.12). Previously, this factorization was only established using strong additional assumptions on the realizability of representations (cf. [Ge1], (2.4), (5.3)).

Let H be an algebra over a local integrally closed domain O with residue field k. Then we have a canonical map from central functions on H to central functions on kH (induced by reduction modulo the maximal ideal of O). In Proposition 3.1, generalizing a theorem of Hattori, we show that the surjectivity of this map implies that the decomposition map has finite cokernel and that the Cartan matrix of kH has non-zero determinant. For a symmetric algebra, this surjectivity is equivalent to the surjectivity of the reduction map Z(H) → Z(kH). In Theorem 5.2 we prove that this surjectivity holds for Iwahori-Hecke algebras, by constructing a basis of the center from [Ge-Pf]. This part is inspired by the work of [Di-Ja], where the type A_{n} was considered. We believe that this stability of the center under reduction is an important similarity between group algebras and Iwahori-Hecke algebras.

This paper is a contribution to the DFG research project on “Algorithmic Number Theory and Algebra”. The second author thanks the Lehrstuhl D für Mathematik, RWTH Aachen, for its hospitality and support during part of this work.

In another direction we show that, under suitable hypothesis, the number of simple modules of the algebra kH is “generic”, in the following sense. Assume that the ground ring O has Krull dimension 2. Fix a height 1 prime ideal 𝔭 and let k_{𝔭} be the quotient field of O/𝔭. Then Theorem 3.3 gives a condition on 𝔭 which implies that the number of simple modules of k_{𝔭}H equals the number of simple modules of kH. In our applications to Iwahori-Hecke algebras, O will be the localization of the ring of Laurent polynomials over ℤ in one indeterminate. The choice of height 1 and height 2 prime ideals yield algebras kH and k_{𝔭}H, where k is a finite field of characteristic ℓ and k_{𝔭} is a cyclotomic field of characteristic 0 (see [Ge1] for more details). In Theorem 5.4 we check that the above hypotheses are satisfied whenever the prime ℓ is not too small (e.g., does not divide the order of the underlying finite Weyl group). Hence the number of simple modules of kH is determined by the algebra k_{𝔭}H, i.e., it is “generic”. This is one step in an attempt to prove the more general conjecture of [Ge1], (5.6), that even the decomposition maps themselves are “generic”.

2. Decomposition maps

It is the purpose of this section to develop the basic theory of decomposition maps for algebras over integrally closed rings. Much of what follows is inspired by [Bra-Ne] and [Se].

2.1. Grothendieck groups and bilinear forms. Let O be a commutative local ring and H an O-algebra, finitely generated and free as an O-module. We denote by K_{0}(H) the Grothendieck group of the category of finitely generated projective left Hmodules and by R_{0}(H) the Grothendieck group of the category of finitely generated H-modules which are free as O-modules (such modules are called H-lattices). The imbedding of the first category into the second one induces a map (“Cartan map”) c_{O} : K_{0}(H) → R_{0}(H). We denote by R_{0}^{+}(H) the subset of R_{0}(H) given by the classes of the H-lattices. Note that R_{0}^{+}(H) generates R_{0}(H).

In what follows, all modules are supposed to be finitely generated.

There is a bilinear form ⟨·, ·⟩_{O} : K_{0}(H) × R_{0}(H) → ℤ defined by

⟨[P], [V]⟩_{O} = rank_{O} Hom_{H}(P, V)

for P a projective H-module and V an H-lattice (where [P] and [V] denote the classes of P and V in K_{0}(H) and R_{0}(H) respectively). The fact that Hom_{H}(P, V) is free over O follows from the existence of an integer n such that P |H^{n}, since then Hom_{H}(P, V) is a direct summand of Hom_{H}(H^{n}, V) ≃ V^{n} as an O-module. Let us now prove that this form is well defined. If [P] = [P_{1}] + [P_{2}], then Hom_{H}(P, V) ≃ Hom_{H}(P_{1}, V) ⊕ Hom_{H}(P_{2}, V). If 0 → V_{1} → V → V_{2} → 0 is an exact sequence of H-lattices, then 0 → Hom_{H}(P, V_{1}) → Hom_{H}(P, V) → Hom_{H}(P, V_{2}) → 0 is exact because P is projective. These two facts show that ⟨·, ·⟩_{O} is indeed well defined.

Let us denote by CF(H) the module of class functions, Hom_{O}(H/[H, H], O) (where [H, H] denotes the O-submodule of H generated by the commutators hh′ − h′h, h, h′ ∈ H). We introduce now a bilinear form (·, ·)_{O} : K_{0}(H) × CF(H) → O as follows :

Let P be a projective H-module. There exists an integer n such that P is a direct summand of H^{n}. Let e be the corresponding idempotent in End_{H}(H^{n}). The latter space can be canonically identified with the space M_{n}(H) of n × n-matrices over H. Let Tr(e) ∈ H be the trace of e. It is straighforward to check that the image of Tr(e) in H/[H, H] depends only on the class [P] of P in K_{0}(H) and that the corresponding map K_{0}(H) → H/[H, H] is additive. If f ∈ CF(H) then we define ([P], f)_{O} = f(Tr(e)).

Assume that f is the character ch([V]) of an H-lattice V (where we denote by ch : R_{0}(H) → CF(H) the character map). One has Hom_{H}(eH^{n}, V) ≃ eV^{n} as Omodules, hence ⟨[P], [V]⟩ = rank_{O} eV^{n}. But ([P], f)_{O} = f(Tr(e)) = 1_{O} · rank_{O} eV^{n}, hence ([P], f) = ⟨[P], [V]⟩ · 1_{O}. This proves the following :

Lemma 2.1. One has (x, ch(y)) = ⟨x, y⟩ · 1_{O} for x ∈ K_{0}(H) and y ∈ R_{0}(H).

We define a semi-group morphism ρ_{O} from R_{0}^{+}(H) to the set Maps(H, O[X]) of maps H → O[X] as follows :

Given an H-lattice M and x ∈ H, we let ρ_{O}([M])(x) be the characteristic polynomial of x acting on the free O-module M.

Let B be a commutative local O-algebra given by t_{B} : O → B. If M is an Omodule, we denote by BM the B-module B ⊗_{O} M. Without specification, tensor products are taken over O, i.e., B ⊗ M means B ⊗_{O} M. There are canonical maps t_{B}^{K_{0}} : K_{0}(H) → K_{0}(BH), t_{B}^{R_{0}} : R_{0}(H) → R_{0}(BH), t_{B}^{CF} : CF(H) → CF (BH) and t_{B}^{M} : Maps(H, O[X]) → Maps(BH, B[X]) induced by extension of scalars. The following lemma gives the compatiblities with extension of scalars :

Lemma 2.2. Let x ∈ K_{0}(H), y ∈ R_{0}(H) and f ∈ CF (H). Then,

t_{B}((x, f)_{O}) = (t_{B}(x), t_{B}(f))_{B},
⟨x, y⟩_{O} = ⟨t_{B}(x), t_{B}(y)⟩_{B},
t_{B}^{R_{0}} ∘ c_{O} = c_{B} ∘ t_{B}^{K_{0}},
t_{B}^{CF} ∘ ch = ch ∘t_{B}^{R_{0}} and
ρ_{B} ∘ t_{B}^{R_{0}} = t_{B}^{M} ∘ ρ_{O}.

Proof. Only the second assertion does not follow directly from the definitions. If x = [P] and y = [V], then ⟨t_{B}(x), t_{B}(y)⟩_{B} = rank_{B}(eV^{n}) ⊗_{O} B, where n is such that P |H^{n} and e is the corresponding idempotent of End_{H}(H^{n}). Since eV^{n} is a free Omodule, one has rank_{B}(eV^{n})⊗_{O} B = rank_{O} eV^{n}, hence ⟨t_{B}(x), t_{B}(y)⟩_{B} = ⟨x, y⟩_{O}.

Lemma 2.3. Assume B is flat over O. Then, the map 1_{B} ⊗ t_{B}^{CF} : B ⊗ CF(H) → CF(BH) is an isomorphism.

Proof. From the exact sequence 0 → [H, H] → H → H/[H, H] → 0, one gets the exact sequence 0 → [BH, BH] → BH → B ⊗ (H/[H, H]) → 0. Hence, B ⊗ (H/[H, H]) ≃ BH/[BH, BH] and finally Hom_{B}(BH/[BH, BH], B) ≃ B ⊗ Hom_{O}(H/[H, H], O).

2.2. Algebras over a field. Let us first recall without proof some classical results about simple algebras (cf [Bki1]).

Assume that O = K is a field.

We have the following commutative diagram :

[figure: H − mod; R_{0}^{+}(H); ρ; Maps(H, K[X]); (H/J(H)) − mod; R_{0}^{+}(H/J(H)); ≃; ρ; Maps(H/J(H), K[X])]

where J(H) denotes the radical of H. Hence, in order to study R_{0}^{+}(H) and its image in Maps(H, K[X]), we can assume that J(H) = 0.

Now, the algebra H is semisimple, i.e., is isomorphic to a finite direct product of simple algebras. So, let us assume that H is simple. Let V be a simple H-module, D = End_{H}(V) and n = dim_{D}(V). Then, H is isomorphic to the ring M_{n}(D^{∘}) of (n × n)-matrices over the skewfield D^{∘} opposite to D.

Let m be the integer such that [D : Z(D)] = m^{2}. Let Trd : H → Z(D) be the reduced trace of the central simple Z(D)-algebra H. It has the property that if L is a neutralizing field for H, i.e., such that Z(D) ⊂ L ⊂ D and H ⊗_{Z(D)} L ≃ M_{mn}(L), then the usual trace M_{mn}(L) → L is given by Trd ⊗1_{L}. Denote by ch_{K}(V) and ch_{Z(D)}(V) the character of V respectively viewed as a module over the K-algebra H and as a module over the Z(D)-algebra H. We have

ch_{K}(V) = Tr_{Z(D)/K} ch_{Z(D)}(V) and ch_{Z(D)}(V) = m Trd

where Tr_{Z(D)/K} : Z(D) → K denotes the trace map of the K-algebra Z(D), i.e., the character of the module Z(D) for the K-algebra Z(D).

Similarly, we have

ρ_{K}(V) = N_{Z(D)[X]/K[X]}(ρ_{Z(D)}(V)) and ρ_{Z(D)}(V) = Prd^{m}

where N_{Z(D)[X]/K[X]} : Z(D)[X] → K[X] is the norm map of the K[X]-algebra Z(D)[X] and where Prd : H → Z(D)[X] is the reduced characteristic polynomial map and ρ_{K}(V) and ρ_{Z(D)}(V) are the characteristic polynomial maps of V respectively viewed as a module over the K-algebra H and as a module over the Z(D)- algebra H.

Lemma 2.4. The following statements are equivalent for H a simple K-algebra with simple module V and D = End_{H}(V) :

(1) the extension Z(D) of K is separable,

(2) Tr_{Z(D)/K} ≠ 0,

(3) Tr_{Z(D)/K} Trd ≠ 0,

(4) the algebra H ⊗_{K} Z(D) is semisimple,

(5) the algebra D ⊗_{K} Z(D) is semisimple,

(6) the H ⊗_{K} Z(D)-module V ⊗_{K} Z(D) is semisimple. If Z(D) is a separable extension of K, then the algebra H ⊗_{K} Z(D) is isomorphic to a direct product of [Z(D) : K] central simple Z(D)-algebras and the module V ⊗_{K} Z(D) is isomorphic to the direct sum of [Z(D) : K] non isomorphic simple modules.

Let us go back to the case where H is any finite dimensional algebra over K. From now, we assume that K is perfect (one could as well work with the weaker

assumption that given any simple H-module V, then Z(End_{H}(V)) is a separable extension of K).

A basis for R_{0}(H) (resp. K_{0}(H)) is given by the images of a complete set of representatives of isomorphism classes of simple (resp. projective indecomposable) modules. Hence, one has an isomorphism R_{0}(H) → K_{0}(H) given by sending the class of a simple module to the class of one of its projective covers.

An irreducible character of H is defined as the character of a simple H-module and we denote by Irr(H) the set of irreducible characters of H.

If K′ is a field extension of K, then the canonical maps t_{K′}^{K_{0}} : K_{0}(H) → K_{0}(K′H) and t_{K′}^{R_{0}} : R_{0}(H) → R_{0}(K′H) are injective.

There exists a finite Galois extension K′ of K such that K′H is split, i.e., such that for every simple K′H-module V, the canonical map K′ → End_{K′H}(V) given by multiplication is an isomorphism ; we call such a field K′ a neutralizing field for H. Let V be a simple H-module. Then, there are non-isomorphic simple K′H-modules V_{1}, …, V_{s} and an integer m_{V} (the Schur index of V) such that

K′V ≃ (V_{1} ⊕ ⋯ ⊕ V_{s})^{m_{V}} .

Note that we have [K_{V} : Z(K_{V})] = m_{V}^{2} where K_{V} = End_{H}(V). Let V′ be another simple H-module, with K′V′ ≃ (V′_{1} ⊕ ⋯ ⊕ V′_{s′})^{m_{V′}} where the modules V′_{i} are simple and V′_{i} ≄ V′_{j} for i ≠ j. Then, if V ≄ V′, we have V_{i} ≄ V′_{j}, for all i, j. Let P_{V} be a projective cover of V. Then, Hom_{K′H}(K′P_{V} , K′V′) ≃ K′ ⊗ Hom_{H}(P_{V} , V′) ≃ K′⊗Hom_{H}(V, V′) ≃ Hom_{K′H}(K′V, K′V′). Hence, K′P_{V} is a projective cover of K′V, i.e., denoting by P_{i} a projective cover of V_{i}, we have

K′P_{V} ≃ (P_{1} ⊕ ⋯ ⊕ P_{s})^{m_{V}} .

We then define R̄_{0}(H) as the subgroup of R_{0}(K′H) with basis \{\frac{1}{m_{V}} [V]\} where V runs over the simple H-modules (cf [Se, §12.1]). Similarly, we define K̄_{0}(H) as the subgroup of K_{0}(K′H) with basis \{\frac{1}{m_{V}} [P_{V}]\} where V runs over the simple H-modules. Note that the group R_{0}(H) (resp. K_{0}(H)) is a subgroup of finite index of R̄_{0}(H) (resp. K̄_{0}(H)). In particular, rank R_{0}(H) = rank R̄_{0}(H) and rank K_{0}(H) = rank K̄_{0}(H).

It is clear that c_{K}, ⟨·, ·⟩_{K} and (·, ·)_{K} extend to maps K̄_{0}(H) → R̄_{0}(H), K̄_{0}(H) × R̄_{0}(H) → ℤ and K̄_{0}(H) × CF(H) → K compatible with the extension to K′.

Furthermore, we define R̄_{0}^{+}(H) as R̄_{0}(H) ∩ R_{0}^{+}(K′H).

Proposition 2.5. The map ρ_{K} : R̄_{0}^{+}(H) → Maps(H, K[X]) is an injection.

Proof. (cf [Bra-Ne, Lemma 2]) It is enough to prove the lemma in the case KH split, which we assume now. Let M and N be two H-modules such that ρ([M]) = ρ([N]). By replacing M and N by their associated semi-simple modules, one can assume that M and N are semi-simple. Let S be a set of representatives of isomorphism classes of simple H-modules. For V ∈ S, let a_{V} and b_{V} be the multiplicities of V as a composition factor of M and N. Since ch([M]) = ch([N]), one has (a_{V} −b_{V})·1_{K} = 0. If O has characteristic zero, this implies a_{V} = b_{V}, hence [M] = [N]. Otherwise, let p be the characteristic of K, p > 0. One has a_{V} ≡ b_{V} mod p.

Let us assume that ρ is an injection for modules of dimension at most n and assume that M and N have dimension n + 1. If there is a V ∈ S which is a submodule of M and N, then the modules M/V and N/V are isomorphic, since their dimension is less that n. Hence, we may assume that for every V ∈ S, a_{V} = 0 or b_{V} = 0. Let M′ = ⊕_{V} V ^{\frac{a_{V}}{p}} and N′ = ⊕_{V} V ^{\frac{b_{V}}{p}} . Again, M′ and N′ have dimension less than n, hence M′ ≃ N′, i.e., M′ = N′ = 0 and M = N = 0, which gives a contradiction.

Lemma 2.6. The subgroup R̄_{0}(H) of R_{0}(K′H) consists of those elements f such that ρ_{K′}(f)(h) ∈ K[X] for all h ∈ H.

Proof. It follows from the construction of R̄_{0}(H) that, for f ∈ R̄_{0}(H) and h ∈ H, then ρ(f)(h) ∈ K.

Let now f ∈ R_{0}(K′H) such that ρ(f)(h) ∈ K for all h ∈ H. We can clearly assume that H is semisimple, since ρ(f)(h + r) = ρ(f)(h) for h ∈ H and r ∈ J(H). We can also assume that H is simple. Let V_{1}, …, V_{s} be a complete set of representatives of isomorphism classes of simple K′H-modules. Then, the Galois group of K′ over K acts transitively on {ch([V_{i}])}, hence ρ(f) = α ∑_{i=1}^{s} ch([V_{i}]) for some α ∈ K. Since ρ is injective, by Lemma 2.5, we have f = α ∑_{i=1}^{s}[V_{i}] for some integer α and the lemma is proved.

We put \overline{Irr}(H) = \{ch(\frac{1}{m_{V}} [V])\} where V runs over the simple H-modules.

Proposition 2.7. The map 1_{K} ⊗ ch : K ⊗ R̄_{0}(H) → CF(H) is an injection, i.e., the elements of \overline{Irr}(H) are linearly independent.

If H is semi-simple, then the map above is an isomorphism.

Proof. One can assume that H is simple and the proposition follows then from Lemma 2.4.

Assume now that for every simple H-module V, the canonical map K → Z(End_{H}(V)) is an isomorphism, i.e., End_{H}(V) is a central K-algebra : we say that H is a quasicentral K-algebra.

Proposition 2.8. If H is a quasi-central K-algebra, then :

• the form ⟨·, ·⟩ induces a perfect pairing between K̄_{0}(H) and R̄_{0}(H) : if V, V′ are two simple H-modules and P_{V} is a projective cover of V, then we have ⟨\frac{1}{m_{V}} [P_{V}], \frac{1}{m_{V′}} [V′]⟩ = δ_{[V],[V′]} ;

• if K′ is an extension of K, then the maps t_{K′}^{R_{0}} : R̄_{0}(H) → R̄_{0}(K′H) and t_{K′}^{K_{0}} : K̄_{0}(H) → K̄_{0}(K′H) are isomorphisms ;

• if K′ is a neutralizing field for H, then R̄_{0}(H) = R_{0}(K′H) and rank R_{0}(H) = rank R_{0}(K′H).

Proof. We have Hom_{H}(P_{V} , V′) ≃ Hom_{H}(V, V′) and dim_{K} Hom_{H}(V, V′) = m_{V}^{2} δ_{[V],[V′]} so that ⟨[P_{V}], [V′]⟩ = m_{V}^{2} δ_{[V],[V′]}. This implies that the pairing induced by ⟨·, ·⟩ is perfect. The other statements are clear.

There is an easy characterization of quasi-central algebras :

Proposition 2.9. The following statements are equivalent :

(1) The K-algebra H is quasi-central ;

(2) R̄_{0}(H) = R̄_{0}(K′H) for any extension K′ of K ;

(3) rank R_{0}(H) = rank R_{0}(K′H) for any extension K′ of K.

(4) there is a finite Galois extension L of K which is a neutralizing field for H and such that rank R_{0}(H) = rank R_{0}(LH)

Proof. Let L be a finite Galois extension of K which is a neutralizing field for H. Let V be a simple H-module and D = End_{H}(V). Then, V ⊗_{K} L is a direct sum of [Z(D) : K] non isomorphic simple LH-modules. If Z(D) ≠ K, it implies rank R_{0}(H) < rank R_{0}(LH). Hence, (4) ⇒ (1).

By Proposition 2.8, (1) implies (2). Finally, (2) ⟹ (3) ⟹ (4) is clear.

2.3. The Brauer-Cartan square. From now, O is an integrally closed local domain, K its field of fractions and k its residue field. We assume K and k are perfect. Let H be an O-algebra, free and finitely generated as an O-module.

Lemma 2.10. The image of ch : R̄_{0}(KH) → CF(KH) is contained in the Osubmodule CF(H) and the image of ρ_{K} : R̄_{0}^{+}(KH) → Maps(KH, K[X]) is contained in the O-submodule Maps(H, O[X]).

Proof. Let V be a simple K′H-module where K′ is a finite extension of K, with ch(V)(x) ∈ K for all x ∈ KH. Then, for h ∈ H, the characteristic polynomial ρ(V)(h) divides (in K′[X]) the characteristic polynomial ρ(KH)(h) = ρ(H)(h) associated to the regular representation. Since ρ(H)(h) ∈ O[X], the roots of ρ(V)(h) are algebraic over O, hence ch(V)(h) is algebraic over O. Since ch(V)(h) ∈ K and O is integrally closed in K, this implies ch(V)(h) ∈ O. Similarly, the coefficients of ρ(V)(h) are algebraic over O and are in K, hence are in O.

The following proposition establishes the existence of the decomposition map :

Proposition 2.11. There exists a unique map d_{O} : R̄_{0}(KH) → R̄_{0}(kH) which makes the following diagram commutative :

[figure: R̄_{0}(KH); d_{O}; R̄_{0}^{+}(KH); ρ_{K}; Maps(H, O[X]); t_{k}; R̄_{0}(kH); R̄_{0}^{+}(kH); ρ_{k}; Maps(kH, k[X])]

Proof. Note first that the unicity follows from the injectivity of ρ_{k} (cf Proposition 2.5).

Let K′ be a finite extension of K such that K′H is split. Let O′ be a valuation ring, O ⊂ O′ ⊂ K′ with maximal ideal I such that I ∩ O = 𝔪, the maximal ideal of O, and residue field k′. Let V be a simple K′H-module with [V] ∈ R̄_{0}(KH). Since finitely generated torsion-free O′-modules are free [Go, §5.2], there exists an O′Hlattice V′ such that K′V′ ≃ V. Then, ρ_{k′}([k′V′]) is the reduction mod I of ρ_{K}([V]) ∈ Maps(O′H, O′[X]). Since ρ_{K}([V]) is actually in Maps(OH, O[X]) by assumption, we have also ρ_{k′}([k′V′]) in Maps(kH, k[X]), hence [k′V′] ∈ R̄_{0}(kH) by Lemma 2.6 and we put d([V]) = [k′V′].

Note that the decomposition map exists not only for O integrally closed, but more generally when the image of ρ_{K} : R̄_{0}^{+}(KH) → Maps(KH, K[X]) is contained in the O-submodule Maps(H, O[X]).

The following proposition is a direct consequence of the definition :

Proposition 2.12. Let 𝔭 be a prime ideal of O such that k_{𝔭} = (O_{𝔭})/𝔭 is perfect and O/𝔭 is integrally closed. Then, the following diagram is commutative :

[figure: R̄_{0}(KH); d_{O𝔭}; d_{O}; R̄_{0}(kH); R̄_{0}(k_{𝔭}H); d_{O/𝔭}]

Suppose now that KH is a quasi-central K-algebra.

We define the map e : K̄_{0}(kH) → K̄_{0}(KH) as the map dual to d with respect to the pairing ⟨·, ·⟩, i.e., for η ∈ K̄_{0}(kH) and χ ∈ R̄_{0}(KH), we have

⟨η, d(χ)⟩_{k} = ⟨e(η), χ⟩_{K}.

Let us now give an alternative definition of e, without using d.

Let Õ be a strict henselisation of O [Ray, Chapitre VIII] : this is an henselian (local) ring, local extension of O, faithfully flat as an O-module, with residue field k̄ a separable closure of k. Furthermore, since O is integrally closed, the ring Õ is an integral domain [Ray, Chapitre IX, corollaire 1]. Let now O′ be a valuation ring, local extension of Õ, contained in the field of fractions K̃ of Õ, with residue field k′.

Since Õ is henselian, every idempotent of k̄H can be lifted to an idempotent of ÕH, hence every projective k̄H-module can be lifted to a projective ÕH-module. Let P be a projective k̄H-module and V a KH-module. There exists an O′H-lattice M such that V ⊗K̃ ≃ M ⊗K̃ and a projective O′H-module Q such that Q⊗k′ ≃ P ⊗k′. We have

⟨[P], d([V])⟩k̄ = ⟨[Q ⊗ k′], [M ⊗ k′]⟩_{k′} = ⟨[Q], [M]⟩_{O′} = ⟨[K̃Q], [K̃M]⟩K̃.

(Note that similarly, ([P], f ⊗ 1k̄)k̄ = (e([P]), f ⊗ 1_{K})_{K} · 1k̄ for f ∈ CF(H)). Hence, e([P]) = [K̃Q] (viewed in K̄_{0}(KH)). Furthermore, one has d ∘ c_{K} ∘ e([P]) = c_{k}([P]), hence, the following diagram

(“Brauer-Cartan square”) is commutative :

[figure: R̄_{0}(KH); d; R̄_{0}(kH); K̄_{0}(KH); c_{K}; K̄_{0}(kH); e; c_{k}]

With the additional assumption that the algebra KH is semi-simple, one has K̄_{0}(KH) = R̄_{0}(KH) and we recover the usual Brauer-Cartan triangle (cf [Se, §15]) :

[figure: R̄_{0}(KH); d; K̄_{0}(kH); e; c_{k}; R̄_{0}(kH)]

3. On the number of simple modules

We keep the assumptions above : We have O an integrally closed local domain with residue field k perfect and field of fractions K perfect and H an O-algebra, free and finitely generated as an O-module.

The following proposition generalizes a theorem of Hattori [CuRe, theorem 32.5] about the injectivity of the Cartan map. In Hattori’s theorem, it is assumed that H/[H, H] is free as an O-module (and O is assumed to be a discrete valuation ring).

Proposition 3.1. Assume kH quasi-central. If the image of the canonical map t_{k} : CF(H) → CF(kH) contains ch(R̄_{0}(kH)), then the map e is injective (hence, given two projective H-modules M and N, we have KM ≃ KN if and only if M ≃ N).

In particular, the decomposition map d has finite cokernel and the algebra KH has at least as many simple modules as the algebra kH.

Proof. Let us prove first that e is injective. Replacing O by a strict henselisation of O, we can assume that kH is split. Let P, Q be two non-zero projective kHmodules with no common direct summand such that e([P] − [Q]) = 0. Assume that P has minimal dimension with this property. Let V be a simple kH-module and put φ = [V]. We have

< [P], φ > ·1_{k} = ([P], ch φ)_{k} = ([P], f · 1_{k})_{k} = (e[P], f)_{K} · 1_{k}.

As e([P]) = e([Q]), we get < [P] − [Q], φ >_{k} ·1_{k} = 0. If k has characteristic zero, it implies that [P] = [Q]. Assume then that the characteristic p of k is positive. Then, the multiplicities of a projective cover P_{V} of V in P and Q as a direct summand are equal modulo p. Since P and Q have no common direct summand by assumption, it implies that the multiplicities of P_{V} in P and Q are both divisible by p.

So, there exists P_{0} and Q_{0} two projective kH-modules with P ≃ P_{0}^{p} and Q ≃ Q_{0}^{p}, hence e([P_{0}] − [Q_{0}]) = 0. Since P_{0} has strictly smaller dimension than P, it implies P_{0} ≃ Q_{0}, hence P ≃ Q, which is impossible.

This completes the proof that ker e = 0.

Now, by definition of e as dual of d, it follows that d has finite cokernel. Since d : R̄_{0}(KH) → R̄_{0}(kH), it is clear that KH has at least as many simple modules as kH.

Assume from now that KH is quasi-central.

Lemma 3.2. Assume the canonical map t_{k} : CF(H) → CF(kH) is surjective. Then, the algebra kH is quasi-central.

Proof. Let k′ be a finite Galois extension of k neutralizing for H. Let O′ be a local domain, local extension of O with residue field k′ and field of fractions K′. By Lemma 2.3, the canonical map CF(O′H) → CF(k′H) is also surjective, hence Proposition 3.1 proves that d_{O′} has finite cokernel.

As KH is quasi-central, one has R̄_{0}(K′H) = R̄_{0}(KH) by Proposition 2.9, hence the image by the decomposition map of R̄_{0}(K′H) is contained in R̄_{0}(kH). Since the decomposition map d_{O′} : R̄_{0}(K′H) → R_{0}(k′H) has finite cokernel, we get rank R_{0}(kH) = rank R_{0}(k′H), hence kH is quasi-central, by Proposition 2.9.

Assume O is noetherian and has Krull dimension 2. Let 𝔭 be a height one prime ideal of O with O/𝔭 integrally closed (i.e., O/𝔭 is a discrete valuation ring) and k_{𝔭} = (O_{𝔭})/𝔭 perfect.

For 𝔮 a height one prime ideal of O, let k̃_{𝔮} be a finite separable extension of k_{𝔮} = (O_{𝔮})/𝔮 neutralizing for k_{𝔮}H and Õ_{𝔮} a discrete valuation ring, (unramified) extension of O_{𝔮} with residue field k̃_{𝔮} and field of fractions K̃_{𝔮} ; let ˆ Õ_{𝔮} denote the

completion of Õ_{𝔮} and ˆ K̃_{𝔮} its field of fractions.

Here is now the crucial result :

Theorem 3.3. Assume the canonical map CF(H) → CF(kH) is surjective. The map 1_{k} ⊗ d_{O/𝔭} : k ⊗ R̄_{0}(k_{𝔭}H) → k ⊗ R̄_{0}(kH) is an isomorphism if and only if

the restriction of the bilinear form (·, ·)_{ˆ_{O}_{𝔭}} to K_{0}( ˆÕ_{𝔭}) × CF(H) has values in O.

If for every height one prime ideal 𝔮 ≠ 𝔭 of O, the algebra k_{𝔮}H is semi-simple, then these two equivalent statements hold. In particular, the number of simple kH-modules is equal to the number of simple k_{𝔭}H-modules.

Proof. Note that k_{𝔭}H is quasi-central since k_{𝔭} is perfect, by Lemma 3.2. Let 𝔮 be a height one prime ideal of O. Note that ˆ Õ_{𝔮} ∩ K = O_{𝔮} (the intersection is taken in

ˆK̃_{𝔮}). Since ˆ Õ_{𝔮} is a complete discrete valuation ring, idempotents can be lifted from k̃_{𝔮}H to ˆ Õ_{𝔮}H. Hence, the canonical map K_{0}( ˆÕ_{𝔮}H) → K_{0}(k̃_{𝔮}H) is an isomorphim.

We have the following commutative diagram :

[figure: CF(H); t_{O/𝔭}; CF(kH); Hom_{k}(kH, k); CF((O/𝔭)H); Hom_{O/𝔭}((O/𝔭)H, O/𝔭)]

We have

CF((O/𝔭)H) = im(t_{O/𝔭}) + (𝔪/𝔭) CF((O/𝔭)H)

(note that CF((O/𝔭)H) is a direct summand of Hom_{O/𝔭}((O/𝔭)H, O/𝔭) as O/𝔭modules since O/𝔭 is a discrete valuation ring), hence im(t_{O/𝔭}) = CF((O/𝔭)H) by Nakayama’s lemma. This proves that the canonical map CF(H) → CF((O/𝔭)H) is surjective, hence the bilinear form (·, ·)_{ˆ_{O}_{𝔭}} restricted to K_{0}( ˆÕ_{𝔭}H) × CF(H) has values

in O if and only if (·, ·)_{k_{𝔭}} restricted to K̄_{0}(k_{𝔭}H) × CF((O/𝔭)H) has values in O/𝔭.

Since ⟨·, ·⟩ induces a perfect pairing between K̄_{0}(k_{𝔭}H) and R̄_{0}(k_{𝔭}H), the submodule ch R̄_{0}(k_{𝔭}H) of CF((O/𝔭)H) is pure if and only if the form (·, ·)_{k_{𝔭}} restricted to K̄_{0}(k_{𝔭}H) × CF((O/𝔭)H) has values in O/𝔭. By Proposition 3.1, the decomposition map d_{O/𝔭} : R̄_{0}(k_{𝔭}H) → R̄_{0}(kH) has finite cokernel, hence the map 1_{k} ⊗ d_{O/𝔭} is an isomorphism if and only if it is injective, i.e., if and only if ch R̄_{0}(k_{𝔭}H) is a pure submodule of CF((O/𝔭)H). Hence, the first part of the theorem is proven.

Assume that for 𝔮 ≠ 𝔭, the algebra k_{𝔮}H is semi-simple. Then, the algebra k̃_{𝔮}H is split semi-simple, hence, the algebra ˆ Õ_{𝔮}H is isomorphic to a direct product of matrix

algebras over ˆ Õ_{𝔮}. So, the canonical map K_{0}( ˆÕ_{𝔮}H) → K_{0}( ˆK̃_{𝔮}H) is an isomorphism

and the form (·, ·)_{ˆ_{K}_{𝔮}} restricted to K_{0}( ˆK̃_{𝔮}H) × CF(H) has values in ˆ Õ_{𝔮} and finally

the form (·, ·)_{K} restricted to K̄_{0}(KH) × CF(H) has values in ∩_{𝔮≠𝔭}O_{𝔮}. Composing

the canonical map K_{0}( ˆÕ_{𝔭}H) → K_{0}( ˆK̃_{𝔭}H) with the inverse of the canonical map

K̄_{0}(KH) → K_{0}( ˆK̃_{𝔭}H) (note that KH is quasi-central), we get that the bilinear form (·, ·)_{ˆ_{O}_{𝔭}} restricted to K_{0}( ˆÕ_{𝔭}H) × CF(H) has values in ⋂_{𝔮≠𝔭} O_{𝔮} ∩ ˆ Õ_{𝔭}, hence in O since O is a Krull ring [Bki2, Chapitre VII, §1, théorème 4].

4. Center and class functions for symmetric algebras

Let O be a commutative ring and H an O-algebra, free and finitely generated as an O-module. Let τ ∈ CF(H). We say that τ is a symmetrizing form for H (cf [Br]) if the induced map

τ̂ : H → Hom_{O}(H, O), h ↦ (h′ ↦ τ(hh′))

is an isomorphism. More concretely, this means that, if B is an O-basis of H then the determinant of the matrix (τ(hh′))_{h,h′∈B} is a unit in O. When such a symmetrizing form exists, we say that the algebra H is symmetric.

Assume now that τ is a symmetrizing form for H. To simplify the notations, for h ∈ H and f ∈ Hom_{O}(H, O), we put h^{∗} = τ̂(h) and f^{∗} = τ̂^{−1}(f). Note that τ̂ induces an isomorphism of O-modules Z(H) → CF(H) : If A is a commutative O-algebra, then the canonical map CF(H) → CF(AH) is surjective if and only if the canonical map Z(H) → Z(AH) is surjective.

If B is an O-basis of H, then the dual basis {b^{∨}}_{b∈B} is defined by the requirement τ(b_{1}b_{2}^{∨}) = δ_{b_{1}b_{2}} for b_{1}, b_{2} ∈ B. For f ∈ Hom_{O}(H, O), one has f^{∗} = ∑_{b∈B} f(b)b^{∨}.

Let M be an H-module. We have a map Tr : End_{O}(M) → End_{H}(M) given by

Tr(f)(m) = ∑_{b∈B} bf(b^{∨}m).

We have Higman’s lemma, following [Br] :

Lemma 4.1. The module M is projective if and only if there is f ∈ End_{O}(M) such that Tr(f) is the identity.

Assume O is an integrally closed integral domain with field of fractions K perfect. Let us assume that KH is quasi-central.

Proposition 4.2. Let χ ∈ \overline{Irr}(KH) and let c_{χ} be the scalar by which χ^{∗} ∈ Z(KH) acts on a simple KH-module V affording a multiple of χ. Then the following hold :

(1) The element c_{χ} lies in O.

(2) The module V is projective if and only if c_{χ} ≠ 0. In particular, the algebra KH is semisimple if and only if c_{χ} ≠ 0 for all χ ∈ \overline{Irr}(KH).

(3) Assume KH semisimple. Then, χ^{∗} = c_{χ}e_{χ} where e_{χ} denotes the central primitive idempotent corresponding to χ (i.e., χ^{∗}e_{χ} ≠ 0). Moreover,

τ = ∑_{χ∈\overline{Irr}(KH)} \frac{1}{c_{χ}} χ.

Proof. Let V be a simple module with character a multiple of χ ∈ \overline{Irr}(KH). The polynomial X − c_{χ} divides ρ(H)(χ^{∗}) : the roots of this polynomial are algebraic over O, hence c_{χ} is algebraic over O. Finally, c_{χ} ∈ K, hence c_{χ} lies in the integral closure of O in K, i.e., in O.

There is a finite extension K′ of K such that K′H is split and it is clear that if the parts (2) and (3) of the proposition hold for K′H, they hold for KH. Hence, we can assume that KH is split.

Let V be a simple KH-module with character χ ∈ Irr(KH). Note that if i is a primitive idempotent of End_{O}(V), then Tr(i) = c_{χ}1_{V}. Hence, if c_{χ} ≠ 0, it follows from Lemma 4.1 that V is projective.

Now, if V is a projective module and e_{χ} the central primitive idempotent of KH such that e_{χ}V ≠ 0, then the algebras e_{χ}KH and (1 − e_{χ})KH are symmetric algebras with symmetrizing form τ_{|e_{χ}KH} and τ_{|(1−e_{χ})KH} and we have KH = e_{χ}KH ⊕ (1 − e_{χ})KH. It is then clear that for the remaining part of the proposition, we can assume that KH is simple and let χ be the unique irreducible character of KH.

Then, Z(KH) = K · 1, hence χ^{∗} = c_{χ}. It implies χ = c_{χ}τ. If i is a primitive idempotent of KH, we have χ(i) = 1, hence c_{χ} ≠ 0 and the proof is complete.

Let now k be the residue field of O which we assume to be perfect.

Proposition 4.3 ([Ge3]). The algebra kH is semisimple if and only if for every χ ∈ \overline{Irr}(KH), we have 1_{k}c_{χ} ≠ 0. If kH is semisimple, then KH is semisimple and kH is quasi-central.

Proof. Suppose kH semisimple. Then, we have dim(kH) ≤ ∑_{V} (\frac{dim V}{m_{V}} )^{2} where V runs over a complete set of representatives of simple kH-modules. Since KH is quasi-central, we have dim(KH) ≥ ∑_{S} (\frac{dim S}{m_{S}} )^{2} where S runs over a complete set of representatives of simple KH-modules. We have d([KH]) = [kH], hence d(∑_{S} [\frac{S}{m_{S}} ]) = ∑_{V} α_{V} [\frac{V}{m_{V}} ] where S (resp. V) runs over a complete set of representatives of simple KH-modules (resp. kH-modules) and α_{V} > 0 for all V. It follows that ∑_{S} (\frac{dim S}{m_{S}} )^{2} ≥ ∑_{V} (\frac{dim V}{m_{V}} )^{2} and we have equality if and only if for all S, then there exists V such that d(S) = d(V) and m_{S} = m_{V}. Now,

∑_{S}(dim S/m_{S})^{2} ≤ dim(KH) = dim kH ≤ (dim V/m_{V})^{2}

hence, we have equalities everywhere above, i.e., kH is quasi-central and KH is semisimple. Futhermore, for χ ∈ \overline{Irr}(KH), then d(χ) ∈ \overline{Irr}(kH). By Proposition 4.2 (2), we get c_{χ}1_{k} = c_{d(χ)} ≠ 0.

Suppose now 1_{k}c_{χ} ≠ 0 for all χ ∈ \overline{Irr}(KH). We have d(χ)(x) = 0 for x ∈ J(kH). Since τ = ∑_{χ} \frac{1}{c_{χ}} χ, we get τ̄(x) = 0 for x ∈ J(kH), where τ̄ = 1_{k} ⊗ τ. Hence, J(kH) is an ideal of kH which is in the kernel of τ̄ : since τ̄ is a symmetrizing form for kH, it implies J(kH) = 0 and kH is semisimple.

Let P be a projective H-module. Let e be an idempotent of M_{n}(H), for some n, such that eH^{n} ≃ P. Let η(P) : Z(H) → O be the restriction to Z(H) of Tr(e)^{∗}. We have η(P)(z) = τ(Tr(e)z) = z^{∗}(Tr(e)) = ([P], z^{∗})_{O}. Note that it implies that η(P) depends only on P.

Assume in addition KH semisimple and split. Given P a projective kH-module, J.Müller suggested considering the map ψ(P) : Z(KH) → K defined by

ψ(P) = ∑_{χ∈Irr(KH)} (e([P]), χ) \frac{1}{c_{χ}} w_{χ}

where w_{χ} is the one dimensional representation of Z(KH) acting (as multiplication by scalars) on a simple KH-module with character χ.

Proposition 4.4. The map ψ(P) restricts to a map Z(H) → O and 1_{k} ⊗ ψ(P) = η(P). In particular,

ψ(P)(1) = ∑_{χ∈Irr(KH)} \frac{(e([P]), χ)}{c_{χ}} ∈ O.

Proof. Let O′ be an henselisation of O : we have K ∩ O′ = O, where the intersection is taken if the field of fractions of O′, since O′ is faithfully flat over O. Hence, to prove the proposition, we can assume that O is henselian. There exists an idempotent e of M_{n}(H) such that keH^{n} ≃ P. Then, η(eH^{n}) is the restriction to Z(H) of Tr(e)^{∗}. We have w_{χ′}(χ^{∗}) = δ_{χ,χ′}c_{χ} for χ, χ′ ∈ Irr(KH), hence η(eH^{n}) = ∑_{χ} \frac{χ(Tr(e))}{c_{χ}}w_{χ} = ψ(P) and 1_{k} ⊗ ψ(P) = 1_{k} ⊗ η(eH^{n}) = η(P).

Note that if H = OG is a group algebra with its usual symmetrizing form, K having characteristic zero and k characteristic p > 0, then the integrality property above is equivalent to the statement that the dimension of a projective kG-module is divisible by the order of a Sylow p-subgroup of G.

When H is an Hecke algebra associated to a Weyl group, with equal parameters and k has characteristic zero, this result was proven in [Ge4, Proposition 2.1] using algebraic groups.

5. Iwahori-Hecke algebras

5.1. Centers of Hecke algebras. We fix a finite Weyl group W with a corresponding set S ⊂ W of simple reflections. Let {u_{s}}_{s∈S} be a set of indeterminates such that u_{s} = u_{t} whenever s, t ∈ S are conjugate in W, and A = ℤ[u_{s}, u_{s}^{−1}]_{s∈S} be the ring of Laurent polynomials in these indeterminates. The generic Iwahori-Hecke algebra H is the A-free A-algebra with A-basis {T_{w}}_{w∈W} and relations :

\begin{cases}T_{w}T_{w′} = T_{ww′} & if l(ww′) = l(w) + l(w′), \\ (T_{s} − u_{s})(T_{s} + 1) = 0 & for s ∈ S \end{cases}

where w ↦ l(w) is the length function on W with respect to the generating set S.

The algebra H is symmetric, with respect to the form τ : H → A defined by τ(T_{1}) = 1 and τ(T_{w}) = 0 for w ≠ 1. The elements in the dual basis of {T_{w}} are given by T_{w}^{∨} = ind(T_{w})^{−1}T_{w^{−1}}, where ind : H → A is the 1-dimensional representation of H defined by ind(T_{s}) = u_{s} for s ∈ S. Thus, we can apply the results of the previous section to the pair (H, τ).

For each conjugacy class C of W, we denote by C_{min} the set of elements of minimal length in C, and we choose one element w_{C} ∈ C_{min}. For each class C and each w ∈ W there exists an element f_{w,C} ∈ A (called class polynomial in [Ge-Pf]) uniquely determined by the property that

φ(T_{w}) = ∑_{C} f_{w,C}φ(T_{w_{C}}) for all φ ∈ CF(H).

(It is shown in [Ge-Pf] that, if w, w′ ∈ C_{min} then T_{w} and T_{w′} are conjugate in H. In particular, every class function of H has the same values on T_{w} and T_{w′}. Hence the definition of f_{w,C} is independent of the choice of w_{C} ∈ C_{min}.)

For each class C we define a function f_{C} : H → A by

f_{C} : T_{w} ↦ f_{w,C} (w ∈ W).

By inverting the defining formula for f_{w,C} above we see that f_{C} is in fact a central function. Hence, given any φ ∈ CF(H), one has

φ = ∑_{C} φ(T_{w_{C}})f_{C},

i.e., the set {f_{C}} is a basis of the A-module CF(H). Using the correspondence between central functions on H and central elements in H we conclude that the elements {z_{C} := f_{C}^{∗}} form an A-basis of the centre of H. Explicitly, we have

z_{C} = ∑_{w∈W} ind(T_{w})^{−1}f_{C}(T_{w})T_{w^{−1}}.

Note that the class polynomials have the following properties. Let C, C′ be conjugacy classes in W, and w ∈ C′. Then

f_{w,C} = δ_{C,C′} if w ∈ C′_{min}.

Moreover, if we specialize all parameters u_{s} to 1 then the function f_{C} specializes to the indicator function of the conjugacy class C and, hence, the element z_{C} specializes to the class sum of C in AW. Thus, the elements {z_{C}} indeed are “generic” analogues of the class sums.

Lemma 5.1. Let B be a commutative A-algebra and z = ∑_{w∈W} a_{w}T_{w} ∈ Z(BH). Let C be a conjugacy class in W. Then the following hold :

(i) If w, w′ ∈ C_{min} then a_{w} = a_{w′}.

(ii) If w ∈ C does not have minimal length, then there exists an element w′ ∈ C and an element s ∈ S such that l(w) = l(w′), l(sw′s) = l(w′) − 2 and a_{w} = a_{w′} = (1/u_{s})a_{sw′s} + (1 − 1/u_{s})a_{sw′}.

Proof. The coefficient a_{w} of T_{w} in z is given by τ(zT_{w}^{∨}). Now assume that T_{v}^{∨} and T_{w}^{∨} (for v, w ∈ W) are conjugate by a unit, say h ∈ BH. Using that τ is a central function and that z is a central element, we deduce that

a_{v} = τ(zT_{v}^{∨}) = τ(zhT_{w}^{∨}h^{−1}) = τ(h^{−1}zhT_{w}^{∨}) = τ(zT_{w}^{∨}) = a_{w}.

Now let w, w′ ∈ C and x ∈ W such that l(w) = l(w′), w′ = xwx^{−1}, and l(xw) = l(x) + l(w). As in [Ge-Pf], we can compute in BH that T_{x}T_{w} = T_{w′}T_{x} hence T_{w} and T_{w′} are conjugate in BH. A similar relation will also hold with w, w′, x replaced by their inverses. Thus, T_{w}^{∨} and T_{w′}^{∨} are conjugate in BH. Using [Ge-Pf, Theorem 1.1], we conclude that T_{w}^{∨} and T_{w′}^{∨} are conjugate in BH, for all w, w′ ∈ C_{min}. Hence, (i) is proved using the above argument.

Now let w, w” ∈ W and s ∈ S such that w” = sws and l(w”) ≤ l(w). As in [Ge-Pf] we see that, if l(w) = l(w”), the element T_{w} is conjugate to T_{sws}. If l(w”) = l(w) − 2 then T_{w} is conjugate to u_{s}T_{sws} + (u_{s} − 1)T_{ws}. Again, similar relations hold with w replaced by w^{−1}. Thus, T_{w}^{∨} will be conjugate either to T_{sws}^{∨} or to (1/u_{s})T_{sws}^{∨} + (1 − 1/u_{s})T_{sw}^{∨}. Now, by [Ge-Pf, Theorem 1.1], there exist w′ ∈ W such that l(w′) = l(w), T_{w′} is conjugate to T_{w} and l(sw′s) = l(w′) − 2. Hence, the argument above implies (ii).

Theorem 5.2. Let B be a commutative A-algebra. Then, the set {1_{B} ⊗ z_{C}} (where C runs over the conjugacy classes of W) forms a B-basis for the centre of BH. In particular, the centre of BH is free as a B-module of rank equal to the number of conjugacy classes in W and the canonical morphism B ⊗ Z(H) → Z(BH) is an isomorphism.

Proof. Since f_{w_{C},C′} = δ_{C,C′}, the elements 1_{B} ⊗ z_{C} are linearly independent in BH. So we must show that they generate Z(BH). The strategy for the following proof is taken from [Di-Ja].

Let z = ∑_{w∈W} a_{w}T_{w} ∈ Z(BH) (where a_{w} ∈ B). Assume that z ≠ 0 and let w ∈ W be of minimal possible length such that a_{w} ≠ 0. Then w lies in some conjugacy class C and we claim that w ∈ C_{min}. This can be seen as follows. Assume, if possible, that w does not have miminal length in C. By Lemma 5.1, there exist some w′ ∈ W and s ∈ S such that a_{w} = a_{w′} = (1/u_{s})a_{sw′s} + (1 − 1/u_{s})a_{sw′}. Since l(w) = l(w′) and l(sw′s) = l(w′) − 2, both sw′s and ws′ have length strictly smaller than w. By the minimality of w, we conclude that a_{sw′s} = a_{sw′} = 0 and, hence, also a_{w} = 0, a contradiction. Thus, a_{w} ≠ 0 for some w ∈ C_{min}. Moreover, Lemma 5.1 (i) shows that a_{w} = a_{w′} for all w′ ∈ C_{min}.

We now consider the element z′ := z − ∑_{C} a_{w_{C}} ind(T_{w_{C}})(1_{B} ⊗ z_{C}) ∈ Z(BH). The above mentioned properties of the elements f_{w,C} show that the coefficient of T_{w} in z′ is zero, for any element w of minimal length in any conjugacy class of W. Thus, z′ = 0 and we are done.

5.2. Number of simple modules for Hecke algebras. Consider the following polynomials associated to irreducible finite Weyl groups :

Q_{A_{n}} = ∏_{i=1}^{n}[i]_{x}

Q_{B_{n}} = ∏_{i=0}^{n−1}[2]_{x^{i}y}(x^{i} + y)[i]_{x}

Q_{D_{n}}= 2[n]_{x} ∏_{i=1}^{n−1}[2i]_{x}
Q_{E_{6}} = 6[2]_{x}[5]_{x}[6]_{x}[8]_{x}[9]_{x}[12]_{x}
Q_{E_{7}} = 6[2]_{x}[6]_{x}[8]_{x}[10]_{x}[12]_{x}[14]_{x}[18]_{x}
Q_{E_{8}} = 30[2]_{x}[8]_{x}[12]_{x}[14]_{x}[18]_{x}[20]_{x}[24]_{x}[30]_{x}
Q_{F_{4}} = 6[6]_{x}[6]_{y}[2]_{xy^{2}}[2]_{x^{2}y}[2]_{xy}[2]_{x^{2}y^{2}}[2]_{x^{3}y^{3}}(x + y^{2})(x^{2} + y)(x + y)(x^{2} + y^{2})(x^{3} + y^{3})
Q_{G_{2}} = 2[2]_{x}[2]_{y}[3]_{xy}(x^{2} + xy + y^{2})

where [i]_{q} = 1 + q + ⋯ + q^{i−1}.

The following proposition gives a criterion for a multi-parameters Iwahori-Hecke algebra to be semi-simple. It generalizes [Gy-Uno] which deals with the equal parameters case and characteristic zero fields.

Assume W is irreducible and let {s_{1}, s_{2}} be a set of representatives of conjugacy classes of elements in S, with s_{1} corresponding to a long root in type B.

Let B = ℤ[\sqrt{u_{s}}, \sqrt{u_{s}}^{−1}]_{s∈S} = A[\sqrt{u_{s}}]_{s∈S} and K = ℚ(\sqrt{u_{s}})_{s∈S} the field of fractions of B. Note that, by [Di-Mi, théorème 3.1], the algebra KH is quasi-central (actually, by [Ge2], the algebra KH is even split, but we won’t need it here).

Proposition 5.3 ([Ge3]). Let k be a perfect field which is a B-algebra. Then, the algebra kH is semi-simple if and only if 1_{k} · Q_{W} (u_{s_{1}}, u_{s_{2}}) ≠ 0.

Proof. A criterion to decide when the specialized Iwahori-Hecke algebra kH is semisimple is given by Proposition 4.3. One has to check that B[\{\frac{1}{c_{χ}} \}_{χ∈\overline{Irr}(KH)}] = B[\frac{1}{Q_{W}}]. Let

us define P_{W} = ∑_{w} ind(T_{w}). For χ an irreducible character of H, we put D_{χ} = \frac{P_{W}}{c_{χ}} : this is the generic degree of χ. The generic degrees are given in [Ca] and one checks easily the property above.

To simplify the exposition, we assume now that there is a set of integers (not all zero) {a_{s}}_{s∈S} (with greatest common divisor 1) such that u_{s} = t^{a_{s}} where t is an indeterminate. In this case, B = ℤ[\sqrt{t}, \sqrt{t}^{−1}], the polynomial Q_{W} is a product of cyclotomic polynomials in t, up to a power of t.

Let ℓ be a prime and q an integer with ℓ |̸q. Let e ≥ 1 be minimal with 1 + q + ⋯ + q^{e−1} = 0 mod ℓ. If d is an integer, we put Φ′_{d} = Φ_{2d}(\sqrt{t}). Let O = B_{𝔪} where 𝔪 is the maximal ideal of B generated by ℓ and by \sqrt{t} − \sqrt{q} if q is a square or by t − q otherwise.

Let I be a prime ideal of O/(Φ′_{e}) containing ℓ. Let k = O/𝔪 ≃ 𝔽_{ℓ}(\sqrt{q}), 𝔭 be the (height one) prime ideal of O generated by Φ′_{e}, O′ = (O/𝔭)_{I} and k′ = (O_{𝔭})/𝔭 the field of fractions of O′. Note that O is integrally closed in K and O′ is integrally closed in k′. Furthermore, the fields K, k′ and k are perfect. As explained above, the algebra KH is quasi-central.

Theorem 5.4. Assume that ℓ and Φ_{eℓ^{r}} do not divide Q_{W} for r > 0. Then, the number of simple kH-modules is equal to the number of simple k′H-modules. More precisely, the map 1_{k} ⊗ d_{O′} : k ⊗ R̄_{0}(k′H) → k ⊗ R̄_{0}(kH) is an isomorphism

Proof. This is a direct consequence of Theorem 3.3 : we need to check its assumptions. The canonical map Z(H) → Z(kH) is an isomorphism by Theorem 5.2, hence the canonical map CF(H) → CF(kH) is surjective, since H is a symmetric algebra (cf §4).

Let now 𝔮 be a height one prime ideal of O, 𝔮 ≠ 𝔭. The algebra ((O_{𝔮})/𝔮) H is semisimple if and only if Q_{W} ∉ 𝔮, according to Proposition 5.3. If ℓ ∈ 𝔮, then Q_{W} ∉ 𝔮. If Φ′_{d} ∈ 𝔮, then Φ′_{d} = Φ′_{eℓ^{r}}, for some integer r, r > 0 since 𝔮 ≠ 𝔭. But Φ_{eℓ^{r}} doesn’t divide Q_{W}, hence Φ′_{eℓ^{r}} doesn’t divide Q_{W} (as a polynomial in \sqrt{t}), hence Q_{W} ∉ 𝔮. It follows that for 𝔮 ≠ 𝔭, then Q_{W} ∉ 𝔮, i.e., the algebra ((O_{𝔮})/𝔮)H is semisimple and we are done.

Remark 5.5. The decomposition map d_{O′} is the map denoted d′_{ℓ} in [Ge1]. In [Ge1], it is conjectured that d_{O} : R̄_{0}(k′H) → R̄_{0}(kH) maps actually classes of simple modules to classes of simple modules.

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Meinolf Geck : Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, 52062 Aachen, Germany

E-mail address: mgeck@math.rwth-aachen.de

Raphaël Rouquier : U.F.R. de Mathématiques de l’Université Paris 7 Denis–Diderot et UMR 9994 du CNRS, 2 Place Jussieu, F–75251 Paris Cedex 05, France

E-mail address: rouquier@mathp7.jussieu.fr