Published in J. Fac. Sci. Univ. Tokyo, Sect. IA, 36(2) (1989), 347-362.

EXTENSION GROUPS FOR MODULAR HECKE ALGEBRAS

Marc Cabanes Equipe de la Théorie des Groupes Finis DMI-Ecole Normale Supérieure 45, rue d’Ulm 75005 PARIS FRANCE

Introduction Hecke algebras were introduced as a fundamental tool in the character theory of Chevalley groups :

they are used to deal with characters involved in induced representations from a Borel subgroup up to the full group. Their standard presentation by generators and relations has a formal analogue where, roughly speaking, the cardinality q = p^{a} of the root subgroups is considered as an indeterminate. Specializations where q = 0 (”0-Hecke algebras”) were studied by P.N. Norton ([4]) : they have a commutative semi-simple quotient and one may find an explicit decomposition into principal indecomposable ideals. At the same time, Sawada introduced ([5]) the algebra H_{k} := End_{kG}Ind_{U}^{G}k, an analogue where the Borel subgroup is replaced by a Sylow p-subgroup U and k is of characteristic p. A decompositon of H_{k} gives rise to a remarkable bijection between simple modules for H_{k} and for kG, leading to a good description of the indecomposable summands of the induced module Ind_{U}^{G}k.

In the present paper we study the extension groups Ext_{H_{k}}^{1}(S, S′) for all simple H_{k}-modules S, S′. Since simple H_{k}-modules are one-dimensional, this amounts to a study of representations of dimension 2 (see 4.). We use a presentation (2.) of H_{k} which is very similar to the one discovered by Yokonuma ([8]) in the case of Chevalley groups. Another ingredient is a decomposition (3.) of the radical J(H_{k}) that leads to a clear dichotomy for Ext^{1} groups (Theorem 16). Some internal consequences for modular and 0-Hecke algebras are given (5. 6.). But the main applications of this work are for kG-modules, as it was the case in Sawada’s work. This will be published later^{∗}.

Notation Through the whole paper we fix p a prime, k an algebraically closed field of characteristic p, G a finite

group with a split BN-pair of characteristic p.

The subgroup B of G satisfies B = U>◁ T = N_{G}(U) where U is a Sylow p-subgroup of G, T is an abelian p′-group. We denote by W = N/T the Weyl group of G, R its set of generators, Φ the (non crystallographic)

∗ ”A criterion for complete reducibility and some applications”, Actes du colloque sur les représentations des groupes finis (Luminy 1988), to appear in Astérisque (1989). root system of basis Δ and same Weyl group : we denote by α ↦ r_{α} ∈ R the associated indexation. Refering to the Coxeter graph of W, we say r and s in R are connected if, and only if, they do not commute. We denote this by r—s. If I and J are subsets of R, we denote their boolean sum (I \ J) ∪ (J \ I) by I + J.

If α ∈ Φ, we denote by X_{α} the associated root subgroup. If Σ ⊂ Δ corresponds to S ⊂ R, one gets the parabolic subgroup P_{S} := BW_{S}B = U_{S} >◁ L_{S}, where L_{S} = T. < X_{α}; α ∈ Φ_{Σ} > is the ”Levi subgroup”. One denotes G_{S} =< X_{α}; α ∈ Φ_{Σ} >. Then the split BN-pair of G endows G_{S} and L_{S} with split BN-pairs of rank |S| by intersection.

We use the standard notations for k-algebras and modules. If A is a k-algebra, A′ a subalgebra, M an A-module, one denotes by Res_{A′}M the restriction of M to A′. The (Jacobson) radical of A is denoted by J(A). If S is a subset of A, one denotes by < S > the two-sided ideal generated by S. If a, b ∈ A, one denotes [a, b] := ab − ba. One defines [A, A] :=< [a, b]; a, b ∈ A >.

1. Modular Hecke algebras. We denote by H_{k}(G) (or simply H_{k}) the endomorphism algebra of the induced kG-module Ind_{U}^{G}k =

kG ⊗_{kU} k, where k stands for the one dimensional trivial kU-module. A k-basis is indexed by N : (a_{n})_{n∈N} defined by a_{n}(1 ⊗ 1) = ∑_{g∈C} g.(n ⊗ 1) where C is a representative system of U/U ∩ nUn^{−1}(see [3]7).

If r ∈ R, one may choose (r) ∈ r ∩ G_{r}, we assume it is the case through the whole paper. We denote T_{r} = T ∩ G_{r}. We also denote T′_{r} :=< t(r)t^{−1}(r)^{−1}; t ∈ T >⊂ T_{r} (see [6]§1). Then the law of H_{k} obeys the following rules (see [3] Theorems 5 and 7(b)) :

(L)  a_{n}a_{n′} = a_{n′n} when l(n′n) = l(n′) + l(n),

a_{n}a_{(r)} = −|T_{r}|^{−1} ∑_{t∈T_{r}} a_{tn} when l((r)n) < l(n),

where l(n) is defined as the length in W of the corresponding class mod. T.

This implies at once that < a_{t}; t ∈ T > is a semi-simple subalgebra isomorphic to kT, we denote it by kT. One also gets

F1. H_{k} is generated as a k-algebra by kT and the a_{(r)}’s for r ∈ R.

2. The standard presentation. In order to construct certain representations of H_{k} (see 4), we need a presentation. We use the relations

first introduced by Yokonuma ([8]), slightly adapted to avoid reference to Chevalley groups. Let us recall we already chose for every r ∈ R a representative in G_{r}.

Definition 1. If r, s ∈ R and rs is of order m, let t_{rs} ∈ G defined by … (s)(r) = … (r)(s)t_{rs} with m terms on the left and m + 1 on the right.

Lemma 2. t_{rs} ∈ T_{r} ∩ T_{s}.

Proof. Standard, see for instance [6] p.141.

The presentation of H_{k} is the following (compare [8] Théorème 3) :

Proposition 3. H_{k} is generated by the generators {a_{t}; t ∈ T } ∪ {a_{(r)}; r ∈ R} subject to the relations : (Y0) a_{t}a_{t′} = a_{t′t} (Y1) a_{t}a_{(r)} = a_{(r)}a_{(r)t(r)^{−1}} (Y2) a_{(r)}^{2} = −|T_{r}|^{−1} ∑_{t∈T_{r}} a_{t}a_{(r)} (Y3_{rs}) a_{(r)}a_{(s)} … = a_{t_{rs}}a_{(s)}a_{(r)} …

Proof. The proof is standard, we restate the main ideas for the convenience of the reader.

Denote by A the algebra defined above. The law (L) implies H_{k} is a quotient. So let us check dim_{k}A ≤ |N|. Consider the monoid of subsets of A with multiplication AA′ = {aa′; a ∈ A, a′ ∈ A′}. If w ∈ W has reduced expression w = r_{1} … r_{l} let A_{w} := {a_{(r_{l})} … a_{(r_{1})}a_{t}; t ∈ T }. By Y1, this is A_{r_{l}} … A_{r_{1}}. It only depends on w : use Y3 and Iwahori-Matsumoto ”word lemma” ([2]§1 Proposition 5). Then Y2 implies the k-span of ∪_{w∈W} A_{w} is an ideal, hence equals A. Each A_{w} has cardinality ≤ |T |, so dim_{k}A ≤ |N|. Remark. It is indeed possible to choose the representatives (r) so that all the t_{rs}’s are 1. This is clear for (untwisted) Chevalley groups (see [8]). This is also proved in general by Tits to be a consequence of a formal analogue of Chevalley’s commutator formula ([7] Proposition 2.9). We thank Professor Tits for having pointed this to us. The commutator formula (P1 in [7]) might be included in the axioms of split BN-pairs since the notion is devised mainly to study Chevalley groups and their twisted analogues (who clearly satisfy it), moreover those are the only split BN-pairs thanks to Fong-Seitz classification.

However, Lemma 2 will be enough for our purpose.

3. The radical. The irreducible representations of H_{k} were classified by Sawada (see [8]). They are given by the following

(see also 6.) :

Definition. If χ ∈ Mor(T, k^{∗}) one denotes R(χ) := {r ∈ R; χ(T_{r}) = 1}. An ”admissible pair” is any pair (χ, J) with J ⊂ R(χ).

Proposition 4. (Sawada) The irreducible representations of H_{k} are one-dimensional. They are under the following form. If (χ, J) is an admissible pair, there is a unique morphism

ψ(χ, J) : H_{k} ⟶ k^{∗},

such that ∀t ∈ T, ψ(χ, J)(a_{t}) = χ(t) and, ψ(χ, J)(a_{(r)}) is −1 if r ∈ J, 0 if r ∈ R \ J.

If I ⊂ R, we consider the subalgebras corresponding to Levi subgroups and G_{I}. By the law in H_{k}, it is clear that ⊕_{n∈N∩G_{I}} ka_{n} ⊂ ⊕_{n∈N∩L_{I}} ka_{n} are subalgebras isomorphic to H_{k}(G_{I}) and H_{k}(L_{I}) respectively. Since the irreducible representations of H_{k}(G) are one-dimensional, they remain irreducible when restricted to any subalgebra, hence

F2. If I ⊂ J ⊂ R then J(H_{k}(G_{I})) ⊂ J(H_{k}(G_{J})) ⊂ J(H_{k}(L_{J})) ⊂ J(H_{k}(G)).

The one-dimensionality of the irreducible representations of H_{k} also implies that H_{k}/J(H_{k}) is isomorphic to some power of k, hence commutative. So

F3. [H_{k}, H_{k}] ⊂ J(H_{k}).

We now give a generation property of J(H_{k}).

Theorem 5. If G is a finite group with a split BN-pair of characteristic p, the radical of the modular Hecke algebra is given by the following : J(H_{k}) = [H_{k}, H_{k}]+ < J(H_{k}(L_{r})); r ∈ R >=< J(H_{k}(G_{rs})); r, s ∈ R such that r—s or r = s > .

The following corollary is an immediate consequence :

Corollary 6. If M is a H_{k}-module, then M is semi-simple if, and only if, ∀r, s ∈ R such that r—s or r = s the restriction Res_{H_{k}(G_{rs})}M is semi-simple.

Proof of the Theorem. In view of F2 and F3, we just have to check J(H_{k}) ⊂ [H_{k}, H_{k}]+ < J(H_{k}(G_{r})); r ∈ R > and J(H_{k}) ⊂< J(H_{k}(G_{rs})); r, s ∈ R such that r—s or r = s >.

Let π : H_{k} → H_{k}/([H_{k}, H_{k}]+ < J(H_{k}(L_{r})); r ∈ R >) be the canonical surjection. Then, by F1, π(H_{k}) is generated by the semisimple subalgebras π(kT) and π(H_{k}(G_{r})) for r ∈ R. On the other hand, π(H_{k}) is commutative. So it is semi-simple (see for instance [1]§7), hence J(H_{k}) = [H_{k}, H_{k}]+ < J(H_{k}(L_{r})); r ∈ R >.

In view of the above, there just remains to check [H_{k}, H_{k}] ⊂< J(H_{k}(G_{rs})); r, s ∈ R r—s or r = s >, or equivalently that A := H_{k}/ < J(H_{k}(G_{rs})); r, s ∈ R r—s or r = s > is commutative. By F1, A is generated by the images of the a_{t}’s and the a_{(r)}’s. The a_{t}’s commute. By F3, if r, s ∈ R, then [a_{(r)}, a_{(s)}] ∈ J(H_{k}(G_{rs})), so the images commute when r—s. If r–/–s then [a_{(r)}, a_{(s)}] = (a_{t_{rs}} − 1)a_{(s)}a_{(r)} by Y 3_{rs}, but t_{rs} ∈ T_{s}. It is clear by Proposition 4 that any irreducible representation ψ(χ, J) of H_{k}(G_{r}) kills (a_{t_{rs}} − 1)a_{(s)}, thus the images of a_{(r)} and a_{(s)} commute. Let us take t ∈ T and r ∈ R, then, by Y1, [a_{(r)}, a_{t}] = a_{t}a_{(r)}(a_{t′} − 1) where t′ = [t, (r)] ∈ T_{r}. Again a_{(r)}(a_{t′} − 1) is in the radical of H_{k}(G_{r}), so the images of a_{t} and a_{(r)} commute. This finishes the proof of the theorem.

The following is stated for future reference. It is a refinement in a particular case.

Proposition 7. If M is an H_{k}-module with all composition factors isomorphic, then M is semi-simple if, and only if, ∀r ∈ R, Res_{H_{k}(G_{r})}M is semi-simple.

Proof. The ”only if” part is clear by F2. We assume ∀r ∈ R, Res_{H_{k}(G_{r})}M is semi-simple. Then each of these restrictions is a sum of isomorphic lines since a Jordan-Hölder sequence for H_{k} remains Jordan-Hölder when restricted. So H_{k}(G_{r}) acts by scalar matrices on M. On the other hand kT is commutative semi-simple. Hence, by F1, there is a decomposition of M as a sum of H_{k}-stable lines. This proves the proposition.

4. Extension groups In order to compute all groups Ext_{H_{k}}^{1}(S, S′) for S, S′ simple (1-dimensional) H_{k}-modules, we study

2-dimensional H_{k}-modules. Thus we have to find almost all subsets of Mat_{2}(k) satisfying the relations of Proposition 3. A preliminary lemma rules the case when G is of rank 1.

Lemma 8. Let (χ, J), (χ′, J′) be two admissible pairs for G. Let r ∈ R and denote ϵ = ψ(χ, J)(a_{(r)}), ϵ′ = ψ(χ′, J′)(a_{(r)}). Let λ ∈ k. Then

a_{t} ↦ ( \begin{matrix}χ(t) & 0 \\ 0 & χ′(t) \end{matrix} ) , a_{(r)} ↦ ( \begin{matrix}ϵ & λ \\ 0 & ϵ′ \end{matrix} )

defines a morphism from H_{k}(L_{r}) to Mat_{2}(k) if, and only if, either λ = 0, or λ ≠ 0 and one of the following is satisfied :

i) (ϵ, ϵ′) = (0, 0), χ′ = χ^{r} and r ∉ R(χ), ii) (ϵ, ϵ′) = (0, −1) or (−1, 0), and χ′ = χ. In particular (ϵ, ϵ′) ≠ (−1, −1) if λ ≠ 0.

Proof. The map defined above is a morphism if, and only if, the matrices satisfy relations Y0-1-2 of Proposition 3. Y0 is clear. The pairs being admissible, the diagonal elements satisfy Y1-2. So Y1-2 amount to the equalities χ′ = χ^{r} and ϵ + ϵ′ = −(Res_{T_{r}}χ, 1) (usual scalar product). Case (ϵ, ϵ′) = (0, 0) is then clear. Case (ϵ, ϵ′) = (−1, −1) is impossible since the scalar product is 0 or 1. In case (ϵ, ϵ′) = (0, −1), we have r ∈ R(χ′), so χ^{′r} = χ′ since T′_{r} ⊂ T_{r} (see 1.). Then the condition is equivalent to χ′ = χ. This completes the proof.

Lemma 9. If

\begin{matrix}ψ : H_{k} & → Mat_{2}(k), \\ h & ↦ ( \begin{matrix}ψ(φ, I)(h) & ψ_{12}(h) \\ 0 & ψ(χ, J)(h) \end{matrix} ) \end{matrix}

is a morphism and u, v ∈ I \ J or u, v ∈ J \ I, then ψ(a_{(u)}) = ψ(a_{(v)}).

Proof. If φ ≠ χ, then Lemma 8 implies ψ_{12}(a_{(u)}) = ψ_{12}(a_{(v)}) = 0, so ψ(a_{(u)}) = ψ(a_{(v)}). Let us assume φ = χ.

If u, v ∈ J \I ⊂ R(χ), we have χ(t_{uv}) = 1, ψ(a_{(u)}) = ( \begin{matrix}0 & ψ_{12}(a_{(u)}) \\ 0 & −1 \end{matrix} ) and ψ(a_{(v)}) = ( \begin{matrix}0 & ψ_{12}(a_{(v)}) \\ 0 & −1 \end{matrix} ). Then

Y3_{uv} gives (−1)^{|uv|−1}ψ(a_{(u)}) = (−1)^{|uv|−1}ψ(a_{(v)}). The same if u, v ∈ I \ J.

In the next three sections we take two admissible pairs (φ, I) and (χ, J), and study the possible extensions corresponding to the associated simple modules. The decomposition J(H_{k}) = [H_{k}, H_{k}]+ < J(H_{k}(L_{r}); r ∈ R > will give rise to two cases of non split extensions.

4.1. A first case. Case 1 for the pair ((φ, I), (χ, J)) is when

(1) φ = χ, I \ J ≠ ∅, J \ I ≠ ∅ and ∀r ∈ I \ J, ∀s ∈ J \ I, r—s.

We consider a morphism

\begin{matrix}ψ : H_{k} & → Mat_{2}(k), \\ h & ↦ ( \begin{matrix}ψ(χ, I)(h) & ψ_{12}(h) \\ 0 & ψ(χ, J)(h) \end{matrix} ) \end{matrix}

Then :

Proposition 10. Choose r ∈ I \ J, s ∈ J \ I and λ, μ ∈ k. Then there is exactly one ψ as above satisfying ψ_{12}(a_{(r)}) = λ and ψ_{12}(a_{(s)}) = μ.

Proof. Let us check first existence. We define ψ on generators and check the relations Y0-3. If t ∈ T, let ψ(a_{t}) = χ(t)I_{2}. If u ∈ R \ (I + J), let ψ_{12}(a_{(u)}) = 0, so that ψ(a_{(u)}) = 0 or −I_{2}. If u ∈ I \ J, let

ψ_{12}(a_{(u)}) = λ, so that ψ(a_{(u)}) = ψ(a_{(r)}) = ( \begin{matrix}−1 & λ \\ 0 & 0 \end{matrix} ). If u ∈ J \ I, let ψ(a_{(u)}) = ψ(a_{(s)}) = ( \begin{matrix}0 & μ \\ 0 & −1 \end{matrix} ).

Then each restriction to H_{k}(L_{v}) is a morphism by Lemma 8, so Y0-2 are satisfied. Let us check Y3_{uv} for u, v ∈ R. If ψ(a_{(u)}) or ψ(a_{(v)}) is scalar, or if ψ(a_{(u)}) = ψ(a_{(v)}), then Res_{H_{k}(L_{uv})}ψ is diagonalizable similar

to ( \begin{matrix}ψ(χ, I) & 0 \\ 0 & ψ(χ, J) \end{matrix} ), so Y3 is satisfied. There remains the case when u ∈ I \ J, v ∈ J \ I. Then u—v by

(1). But ψ(a_{(u)})ψ(a_{(v)}) = 0, so we have 0 on both sides of Y3.

We show uniqueness. Let ψ satisfy the hypotheses of the proposition. Then ψ_{12}(kT) = 0 since the corresponding 2-dimensional representation of kT is semi-simple with isomorphic composition factors (χ = φ). If v ∈ R \ (I + J), let us check ψ_{12}(a_{(v)}) = 0. We cannot have both v—r and v—s, since this would provide a cycle v—r—s—v. Suppose v–/–r and apply Y3_{vr}. We have t_{vr} ∈ T_{r} ⊂ kerχ since r ∈ R(χ) (Lemma 2), so [ψ(a_{(v)}), ψ(a_{(r)})] = 0. On the other hand ψ(χ, I)(a_{(v)}) = ψ(χ, J)(a_{(v)}), hence ψ_{12}(a_{(v)}) = 0. There remains to check that if v ∈ I \J, ψ(a_{(v)}) = ψ(a_{(r)}). If v ∈ I \J (resp. v ∈ J \I) then ψ(a_{(v)}) = ψ(a_{(r)}) (resp. ψ(a_{(v)}) = ψ(a_{(s)})) by Lemma 9.

4.2. A second case. Case 2 for the pair ((φ, I), (χ, J)) is when

(2) there is r ∈ R \ R(φ) such that φ = χ^{r}, {s ∈ R; s—r} ⊃ I + J and (|rs| = 3 ⇒ s ∉ I ∩ J).

We consider a morphism

\begin{matrix}ψ : H_{k} & → Mat_{2}(k), \\ h & ↦ ( \begin{matrix}ψ(φ, I)(h) & ψ_{12}(h) \\ 0 & ψ(χ, J)(h) \end{matrix} ) \end{matrix}

Then :

Proposition 11. Suppose I = J. Let R′ be the set of elements s ∈ R \ R(χ) such that φ = χ^{s} and there is no v ∈ I such that |sv| = 3. Then if (λ_{s})_{s∈R′} is a family of elements of k, there exists exactly one ψ such that ∀t ∈ T, ψ_{12}(a_{t}) = 0 and ∀s ∈ R′, ψ_{12}(a_{(s)}) = λ_{s}.

Proposition 12. Suppose |I + J| = 1 and φ ≠ χ. Let R′′ := {s ∈ R \ R(χ); φ = χ^{s} and s—(I + J)}. Then, if (λ_{s})_{s∈R′′} is a family of elements of k, there exists exactly one ψ such that ∀s ∈ R′′, ψ_{12}(a_{(s)}) = λ_{s}.

Proposition 13. Suppose |I + J| = 1 and φ = χ. Keep R′′ as above. Then, if (λ_{s})_{s∈R′′∪(I+J)} is a family of elements of k, there exists exactly one ψ such that ∀t ∈ T, ψ_{12}(a_{t}) = 0 and ∀s ∈ R′′ ∪(I +J), ψ_{12}(a_{(s)}) = λ_{s}.

Proposition 14. Suppose |I + J| > 1 and φ ≠ χ. There is a unique r satisfying (2). If λ ∈ k, there is exactly one ψ such that ψ_{12}(a_{(r)}) = λ and ∀t ∈ T, ψ_{12}(a_{t}) = 0.

Proposition 15. Suppose |I + J| > 1 and φ = χ. There is a unique r satisfying (2). Choose s ∈ I + J. If λ, μ ∈ k, there is exactly one ψ such that ψ_{12}(a_{(r)}) = λ and ψ_{12}(a_{(s)}) = μ.

Proof of Proposition 11. Existence : if v ∈ R \ R′, let ψ_{12}(a_{(v)}) = 0. Then Y0-2 are easy to check by Lemma 8. Let us check Y3_{uv} for all u, v ∈ R. If ψ(a_{(v)}) or ψ(a_{(u)}) = 0, or if ψ(a_{(u)}) = ψ(a_{(v)}) = −I_{2}, it is clear.

In the remaining cases one may assume v ∈ R′, then ψ(a_{(v)}) = ( \begin{matrix}0 & λ_{v} \\ 0 & 0 \end{matrix} ), and ψ(a_{(u)}) = −I_{2} or ( \begin{matrix}0 & λ_{u} \\ 0 & 0 \end{matrix} ).

The second case is trivial : both sides of Y3 are 0. In the first case u ∈ I ⊂ R(χ) ∩ R(φ), so ψ(t_{uv}) =I_{2}. If u–/–v, Y3_{uv} is a commutation, it is satisfied. If u—v then |uv| > 3 and there is zero on both sides of Y3_{uv}.

Uniqueness. Let ψ be a morphism as in Proposition 11, and let us check that ψ_{12}(a_{(v)}) = 0 when v ∈ R \ R′. If ψ_{12}(a_{(v)}) ≠ 0, then Lemma 8 implies v ∈ R \ R(χ) and φ = χ^{v} (case ii) does not occur since

I = J). So, if v ∉ R′ there is some s ∈ I with |vs| = 3. Then in Y3_{sv} there is ψ(a_{(v)}) = ( \begin{matrix}0 & ψ_{12}(a_{(v)}) \\ 0 & 0 \end{matrix} ) on

the left and 0 on the right. This completes the proof of Proposition 11. Other proofs. Existence. Take everywhere ψ_{12}(kT) = 0. For Propositions 12-13, take ψ_{12}(a_{(v)}) = 0 where non specified by the statement. For Proposition 14, take ψ_{12}(a_{(v)}) = 0 if v ≠ r. For Proposition 15, when v ∈ I + J take ψ_{12}(a_{(v)}) = μ if v and s are both in I \ J or J \ I, −μ otherwise ; when v ∉ I + J ∪ {r}, take ψ_{12}(a_{(v)}) = 0.

Uniqueness is proved much as for Propositions 10 and 11, using Lemmas 8-9 and Y3.

The uniqueness of r in Propositions 14-15 comes from acyclicity : if r, r′ satisfy (2) and s, s′ are in I +J,

(2) implies r—s—r′—s′—r.

4.3. Extension groups.

We now can prove our main result

Theorem 16. Let H_{k} := End_{kG}Ind_{U}^{G}k be the modular Hecke algebra in characteristic p of a finite group with a split BN-pair of the same characteristic, let ψ(φ, I) and ψ(χ, J) be two irreducible representations of H_{k} (see Proposition 4). The group

Ext_{H_{k}}^{1}(ψ(χ, J), ψ(φ, I))

is non zero if, and only if, one of the following is satisfied :

(1) φ = χ, I \ J ≠ ∅, J \ I ≠ ∅, and ∀r ∈ I \ J, ∀s ∈ J \ I, r—s,

(2) there is r ∈ R \ R(χ) such that φ = χ^{r}, {s ∈ R; s—r} ⊃ I + J (boolean sum) and (|rs| = 3 ⇒ s ∉ I ∩ J), the first case corresponding to non-split extensions that are split when restricted to each H_{k}(L_{r}) for r ∈ R. The dimension of Ext_{H_{k}}^{1}(ψ(χ, J), ψ(φ, I)) is given by the following : it is 1 when |I + J| > 1 (this includes case (1)). In case (2) with I = J it is |{s ∈ R \ R(χ); φ = χ^{s} and ∀v ∈ I |sv| ≠ 3}|. In case (2) with |I + J| = 1, it is |{s ∈ R \ R(χ); φ = χ^{s} and s—(I + J)}|.

Proof. The dimensions are easy to derive from Propositions 10-15. They are clearly non zero. Let us take ψ(χ, J) and ψ(φ, I) two irreducible representations of H_{k}, and assume

\begin{matrix}ψ : H_{k} & → Mat_{2}(k), \\ h & ↦ ( \begin{matrix}ψ(φ, I)(h) & ψ_{12}(h) \\ 0 & ψ(χ, J)(h) \end{matrix} ) \end{matrix}

is a morphism such that ψ(J(H_{k})) ≠ 0. Using the decomposition J(H_{k}) =< J(H_{k}(L_{r})); r ∈ R > +[H_{k}, H_{k}] of Theorem 5, one of the following must happen : (1’) ∀r ∈ R, ψ(J(H_{k}(L_{r}))) = 0, and ψ([H_{k}, H_{k}]) ≠ 0. (2’) There exists r ∈ R such that ψ(J(H_{k}(L_{r}))) ≠ 0.

To complete the proof of the theorem it suffices to show that 1’ implies 1 and 2’ implies 2. Case 1’. Since ∀r ∈ R, ψ([H_{k}(L_{r}), H_{k}(L_{r})]) = 0, we have ∀r ∈ R, t ∈ T, ψ([a_{(r)}, a_{t}]) = 0. So there are r, s ∈ R such that ψ([a_{(r)}, a_{(s)}]) ≠ 0. Then ψ(a_{(r)}), ψ(a_{(s)}) are semi-simple non scalar, so case i) of Lemma 8 does not occur : {r, s} ⊂ I +J. Moreover, case ii) must occur for r or s : otherwise ψ_{12}(a_{(r)}) = ψ_{12}(a_{(s)}) = 0 contradicts non commutation. Thus φ = χ. For the same reason, Lemma 9 implies r and s are not both in I \ J or J \ I. Assume r ∈ I \ J and s ∈ J \ I, then I \ J and J \ I are non empty. There remains to check ∀u ∈ I \ J, u—s. Lemma 9 implies ψ(a_{(u)}) = ψ(a_{(r)}). Since I \ J ⊂ R(χ), χ(t_{us}) = 1. If we had u–/–s, Y3_{us} would say [ψ(a_{(r)}), ψ(a_{(s)})] = [ψ(a_{(u)}), ψ(a_{(s)})] = 0, a contradiction. Case 2’. We have ψ_{12}(a_{(r)}) ≠ 0 and case i) of Lemma 8 is satisfied since case ii) is semi-simple. So

r ∈ R \ R(χ) and φ = χ^{r}. If s ∈ I + J, let us check s—r. We have ψ(a_{(r)}) = ( \begin{matrix}0 & ψ_{12}(a_{(r)}) \\ 0 & 0 \end{matrix} ) and

ψ(a_{(s)}) = ( \begin{matrix}0 & ψ_{12}(a_{(s)}) \\ 0 & −1 \end{matrix} ) or ( \begin{matrix}−1 & ψ_{12}(a_{(s)}) \\ 0 & 0 \end{matrix} ). So ψ(a_{(r)})ψ(a_{(s)}) or ψ(a_{(s)})ψ(a_{(r)}) is zero while the other

is −ψ(a_{(r)}) ≠ 0, so Y3_{rs} implies r—s.

Remark. In case (1), I \ J or J \ I has a single element : if r, r′ ∈ I \ J and s, s′ ∈ J \ I then r—s—r′—s′—r, a cycle.

The reader may have noticed that our convention for ψ(χ, J) is opposite to the classical one : in [5], our ψ(χ, J) would be denoted by ψ(R(χ) \ J, χ). An argument for our convention is that conditions (1) and (2) would then become more complicated.

5. Blocks. We now derive from Theorem 16 some lemmas leading to the determination of the blocks of H_{k}. Let us recall that W acts on Mor(T, k^{∗}) by χ(ntn^{−1}) = χ^{w}(t) = (w^{−1}.χ)(t) when n ∈ w ∈ N/T. Let

(χ, J) be an admissible pair for G.

Lemma 17. If φ ∈ W.χ, then ψ(φ, ∅) is in the block of ψ(χ, ∅).

Lemma 18. If we have r ∈ R \ R(χ) and s ∈ J with r—s, then (φ, I) := (χ^{r}, {s ∈ J; s–/–r}) is admissible and ψ(φ, I) is in the block of ψ(χ, J).

Lemma 19. If we have r ∈ R(χ) \ J and s ∈ J such that r—s, then ψ(χ, J + {r, s}) is in the block of ψ(χ, J).

Proofs. In 17, one may assume φ = χ^{r} for r ∈ R. Then, if r ∈ R(χ), χ^{r} = χ = φ and it is trivial. If r ∈ R \ R(χ), then Theorem 16 (2) applies.

In 18, I ⊂ R(φ) : if s–/–r, r normalizes T_{s} by the action of W on roots, so χ^{r}(T_{s}) = χ(T_{s}) which is 1 if s ∈ J ⊂ R(χ). Then Theorem 16 (2) applies.

In 19, Theorem 16 (1) applies.

Concerning the action of W on Mor(T, k^{∗}), we know that if r ∈ R(χ) then χ^{r} = χ. So, if R(χ) = R, χ is W -fixed ; it is even extendible to a unique χ̂ ∈ Mor(N, k^{∗}) such that ∀r ∈ R χ̂((r)) = 1 : a presentation of N is given by T ∪ {(r); r ∈ R} subject to relations defining T, the action of (r) on T by conjugation, the square of (r) (in T_{r}), and … (r)(s)t_{rs} = … (s)(r) as in Definition 1 with t_{rs} ∈ T_{r} ∩ T_{s} ⊂ ker χ.

Definition 20. M_{0} := {χ; R(χ) = R} ⊂ Mor(T, k^{∗}), M′ := Mor(T, k^{∗}) \ M_{0}. If χ ∈ Mor(T, k^{∗}), let e_{χ} := |T |^{−1} ∑_{t∈T} χ(t^{−1})a_{t}. If χ ∈ M_{0}, let b_{χ}^{∅} := |T |^{−1} ∑_{n∈N} χ̂(n^{−1})a_{n}, b_{χ}^{R} := (−1)^{l(w_{0})}|T |^{−1} ∑_{n∈w_{0}} χ̂(n^{−1})a_{n}, b′_{χ} := e_{χ} − b_{χ}^{∅} − b_{χ}^{R}, where w_{0} ∈ W is the element of maximal length. If X ∈ M′/W (:= the set of W -orbits in M′), let b_{X} := ∑_{χ∈X} e_{χ}.

Theorem 21. Let H_{k} :=End_{kG}Ind_{U}^{G}k be the modular Hecke algebra in characteristic p of a finite group G with irreducible split BN-pair of the same characteristic and rank ≥ 2.

i) There are 2(T : T ∩ G_{R}) + |W |^{−1} ∑_{w∈W} (T : [T, w]) blocks in H_{k}, given by the following primitive idempotents in the center Z(H_{k}) :

{b_{X}; X ∈ M′/W } ∪ {b_{χ}^{∅}; χ ∈ M_{0}} ∪ {b_{χ}^{R}; χ ∈ M_{0}} ∪ {b′_{χ}; χ ∈ M_{0}}.

ii) If (χ, J) is an admissible pair, the corresponding representation ψ(χ, J) is in the block corresponding to : b_{X} if R(χ) ≠ R and χ ∈ X, b_{χ}^{∅} if R(χ) = R and I = ∅, b_{χ}^{R} if R(χ) = R and I = R, b′_{χ} if R(χ) = R and R ≠ I ≠ ∅. iii) If |R| = 1, the number of blocks is \frac{1}{2}(|T | + (T : T′_{r})) + (T : T_{r}), the last set in the union above being empty. The blocks corresponding to the b_{χ}^{∅}’s and the b_{χ}^{R} for χ ∈ M_{0} are simple. The blocks corresponding to the b_{X}’s are uniserial of length two. H_{k} is then uniserial with indecomposable modules of dimensions 1 and 2. iv) If |R| ≥ 2, the simple blocks correspond to the b_{χ}^{∅}’s and the b_{χ}^{R}’s. The other uniserial blocks correspond to the b′_{χ}’s for χ ∈ M_{0} when |R| = 2, their length is \frac{1}{2}|W | − 1.

Proof. The e_{χ}’s are clearly idempotents of kT and, if n ∈ N, e_{χ}a_{n} = a_{n}e_{n.χ} by (L). So the b_{X}’s are central idempotents. On the other hand, if R(χ) = R, (L) implies e_{χ}b_{χ}^{∅} = b_{χ}^{∅}e_{χ} = b_{χ}^{∅}, e_{χ}b_{χ}^{R} = b_{χ}^{R}e_{χ} = b_{χ}^{R}, e_{χ}b′_{χ} = b′_{χ}e_{χ} = b′_{χ}, a_{r}b_{χ}^{∅} = b_{χ}^{∅}a_{r} = 0, a_{r}b_{χ}^{R} = b_{χ}^{R}a_{r} = −b_{χ}^{R} for all r ∈ R. So b_{χ}^{∅}, b_{χ}^{R}, b′_{χ} are central idempotents. The images by an irreducible representation ψ(φ, I) are as follows : ψ(φ, I)(b_{X}) = 1 if, and only if, φ ∈ X ⊂ M′, ψ(φ, I)(b_{χ}^{∅}) = 1 if, and only if, φ = χ and I = ∅, ψ(φ, I)(b_{χ}^{R}) = 1 if, and only if, φ = χ and I = R, ψ(φ, I)(b′_{χ}) = 1 if, and only if, φ = χ and ∅ ≠ I ≠ R.

The cases being disjoint, we have orthogonality. Their number is clearly |Mor(T, k^{∗})/W | + 2|M_{0}| = |W |^{−1} ∑_{w∈W} (T : [T, w]) + 2(T : T_{R}) where T_{R} :=< T_{r}; r ∈ R >, this subgroup equals T ∩ G_{R} by [3] 11.4.

There remains to check primitivity, or equivalently that two arbitrary irreducible representations are in the same block when they send the same idempotent to 1 above.

First case is for a representation ψ(χ, J) with R(χ) ≠ R, let B be its block. We must show that all representations ψ(φ, J) with φ ∈ W.χ are in B. By Lemma 17, it is enough to check it is the case for some ψ(φ, ∅) with φ ∈ W.χ. Let m be the smallest cardinality of sets I such that some ψ(φ, I) is in B, then m ≤ |J| ≤ R(χ) < |R|. Assume m is not zero. Among the ψ(φ, I)’s in B with |I| = m, choose one such that the distance d ≥ 1 of I to R \ R(φ) in the Coxeter diagram is minimal. If d = 1 there is r ∈ R \ R(φ) and s ∈ I such that s—r, then Lemma 18 says ψ(φ^{r}, {s ∈ I; s—r}) is in B, a contradiction with the minimality of m. So d ≥ 2, and there are r_{1}—r_{2}—… r_{d} with r_{1} ∈ R \ R(φ) and r_{d} ∈ I. Then r_{d−1} ∈ R(φ) \ I and Lemma 19 says ψ(φ, I + {r_{d−1}, r_{d}}) is in B. But |I + {r_{d−1}, r_{d}}| = m and the distance to R \ R(φ) is now d − 1, a contradiction.

The second case is when ψ(χ, J) satisfies R(χ) = R and |J| = 1. If B denotes its block, we must show that any representation ψ(χ, I) with R ≠ I ≠ ∅ is in B. If I has just one element, repeated application of Lemma 19 allows to move it to the element of J along the Coxeter graph. If |R| > |I| > 1, Lemma 19 allows to assume there is a ”gap” in I : r—a—s with r ≠ s in I and a ∈ R \ I. Then Ext_{H_{k}}^{1}(ψ(χ, I), ψ(χ, I + {a, r, s})) ≠ 0 by Theorem 16(1), so ψ(χ, I + {a, r, s}) is in B. We have decreased the cardinality of I. Repeating this, we would get down to 1. The proof of i) and ii) is now complete. Adaptations in the case |R| = 1 are clear.

Concerning uniserial and simple blocks, one knows that ψ(χ, J) is in a uniserial (resp. simple) block if, and only if, ∑_{φ,I}dim Ext_{H_{k}}^{1}(ψ(χ, J), ψ(φ, I)) ≤ 1 (resp. = 0). So they are easily derived from Theorem 16. Lengths equal dimensions, so they follow from dim e_{χ}H_{k} = |W | when χ ∈ M_{0}. This finishes the proof of the theorem.

Remark. The computation of blocks can be made by giving explicitly the principal indecomposable modules for H_{k}, as done in [4] for the 0-Hecke algebra H_{k}(G, B) = e_{1}H_{k}. If (χ, J) is admissible and χ̂ is the extension of χ to N ∩ L_{R(χ)} such that χ̂(R(χ)) = 1, let us denote a_{χ,J} := ∑_{n∈N_{R(χ)\J}} χ̂(n^{−1})a_{n}. A slight adaptation of [4] would show that the p.i.m. corresponding to ψ(χ, J) is

U_{χ,J} = ⊕_{w∈X_{R(χ)\J} \X_{J}} k.a_{(w)}a_{χ,J}

where X_{S} := {w ∈ W; ∀s ∈ S l(sw) > l(w)}. As in [4], a filtration is given by taking elements of increasing lengths. The Cartan invariant c(ψ(χ, J), ψ(φ, I)) is then proved to be |{w ∈ (X_{R(χ)\J} \ X_{J}) ∩ (X_{R(φ)\I} \ X_{I})^{−1} ; χ^{w} = φ}|. This may also lead to the determination of blocks, but it is not clear whether one can find the Ext^{1} groups in the same fashion : one needs (two first terms of) all filtrations of U_{χ,J}. Remark. The 0-Hecke algebra H_{k}(G, B) := End_{kG}Ind_{U}^{G}k equals e_{1}H_{k} with e_{1} as above (see [3] 9.4). Since e_{1} is central, irreducible representations and Ext^{1}’s for H_{k}(G, B) are easy to derive from what we know about H_{k}. Irreducible representations are the restrictions of ψ(1, I)’s for I ⊂ R, and Ext_{H_{k}(G,B)}^{1}(ψ(1, I), ψ(1, J)) is non zero if, and only if, I \ J ≠ ∅, J \ I ≠ ∅ and ∀r ∈ I \ J, ∀s ∈ J \ I, r—s. Then its dimension is 1. The blocks would be easy to derive, thus providing another proof of [4] 5.2 (see also next section).

6. Concluding remark : generalized 0-Hecke algebras.

The arguments used in sections 3.4.5. seem a bit more general and might cover other examples of rings associated to a Coxeter system acting on an algebra (replacing the group algebra kT). An instance in arbitrary characteristic would be the ”0-Hecke algebra” of [4], or the specialized version where u = 0 of the generic algebra of [8].

Let us take now K an algebraically closed field, A an algebra over K and (W, R) a Coxeter system. We assume W acts on A by algebra automorphisms : (a ↦ w(a)) ∈Aut A. We assume there is a map

\begin{matrix}z : R & → A \\ r & ↦ z(r). \end{matrix}

Let us define the algebra over K, denoted by H_{A}(W), generated by A ∪ {[r], r ∈ R} subject to : (Y0) a.b = ab when a, b ∈ A, (Y1) a.[r] = [r].r(a), (Y2) [r]^{2} = z(r).[r], (Y3_{rs}) [r].[s] … = [s].[r] … (|rs| terms on each side) for all r, s ∈ R such that rs is of finite order.

The last relation implies that if w = r_{1} … r_{l(w)}, then [w] := [r_{1}] … [r_{l(w)}] only depends on w. Let B be a basis of A over K. Using the same idea as in [2] 23 p.55, one may prove that (b[w])_{b∈B,w∈W} is a basis of H_{A}(W) if, and only if, the following are satisfied : (R) ∀r ∈ R ∀a ∈ A az(r) = z(r)r(a), (W) ∀r, s ∈ R and w ∈ W such that rw = ws and l(rw) > l(w), z(r) = w(z(s)).

Assume R and W are satisfied, then we have the following (compare with [5] 3.5) :

Theorem 22. If W is finite and A is finite dimensional, the simple H_{A}(W)-modules remain simple when restricted to A. If moreover A/J(A) is commutative, the simple H_{A}(W)-modules are in bijection with pairs (χ, I) where χ ∈Mor(A, K) and I ⊂ R_{A}(χ) := {r ∈ R; χ(z(r)) ≠ 0} ; one has r ∈ R_{A}(χ) ⇒ χ = r.χ.

Proof. (see also [5]) Let V be a simple H_{A}(W)-module. Let 0 ≠ S ⊂ Res_{A}V be a submodule of minimal dimension. Let w ∈ W of maximal length such that [w].S ≠ 0. Then [w].S is A-stable by Y1, so it is simple as A-module by minimality of dimension. It suffices to check [w].S is stable under H_{A}(W). Let r ∈ R, if l(rw) > l(w) [r].[w].S = [rw].S = 0 by maximality of l(w), if l(rw) < l(w) [r].[w].S = z(r).[w].S ⊂ [w].S. This completes the proof of the first statement.

If A/J(A) is commutative, each simple A-module is one-dimensional, identified with some χ ∈ Mor(A, K). Moreover, if χ is the restriction of ψ ∈Mor(H_{A}(W), K) then, by Y2, (ψ([r]))^{2} = ψ([r])χ(z(r)). So ψ([r]) is 0 or χ(z(r)), and necessarily 0 if r ∈ R \ R_{A}(χ). Moreover, if r ∈ R_{A}(χ), condition (R) implies χ = r.χ.

Conversely, if χ ∈ Mor(A, K) and I ⊂ R_{A}(χ), there is one ψ ∈Mor(H_{A}(W), K) such that Res_{A}ψ = χ, ψ([r]) = χ(z(r)) if r ∈ I and ψ([r]) = 0 if r ∈ R \ I : it suffices to check Y0-3 are satisfied. Y0-2 are clear. Y3_{rs} is clear when |rs| is even or {r, s} ⊄ I. If r, s ∈ I ⊂ R_{A}(χ) and |rs| is odd, then r and s are conjugate by some w ∈ W_{{r,s}}, so w fixes χ and (W) implies χ(z(s)) = χ(z(r)), hence Y3_{rs} is satisfied in that case.

From this, it would be easy to derive analogues of Theorems 5, 16, 21 for this kind of general ”0-Hecke algebra” H_{A}(W), when A/J(A) is commutative. In Theorem 5, one has to add J(A) which is not necessarily 0. Non semi-simplicity of A causes also a slight change in Theorem 16. The value of dim Ext_{H_{A}(W)}^{1}(ψ(χ, J), ψ(φ, I)) is the same as in Theorem 16 except in one case :

(3) I = J and Ext_{A}^{1}(χ, φ) ≠ 0. In case (3) one proves easily, using the defining relations Y1-3, that dim Ext_{H_{A}(W)}^{1}(ψ(χ, J), ψ(φ, J)) = dim Ext_{A}^{1}(χ, φ) + |{r ∈ R \ R_{A}(χ); φ = r.χ , ∀s ∈ J |sr| ≠ 3}|. The change on blocks is as follows. The group W permutes the blocks of A. Using (R), one proves easily that if R_{A}(χ) = R and Ext_{A}^{1}(χ, φ) ≠ 0, then R_{A}(φ) = R. Then, the blocks of H_{A}(W) are obtained as in Theorem 21, replacing Mor(T, k^{∗}) with the set of blocks of A and M_{0} with the set of blocks of irreducible representations χ of A such that R_{A}(χ) = R.

Some adaptations are also possible to study the two-dimensional representations of H_{A}(W) when W is no longer finite. One must take care of squares and triangles in the Coxeter diagram, since we used repeatedly acyclicity in the discussions of 5. Cycles of larger size do not seem to cause any change.

References

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Math. Z., 155(1977),29-41. [6]. H. Sawada, Endomorphism rings of split (B, N)-pairs, Tokyo J. Math. , 1(1) (1978), 139-148. [7]. J. Tits, Moufang octagons and the Ree groups of type ^{2}F_{4}, Amer. J. Math. , 105 (1983), 539-594. [8]. T. Yokonuma, Sur la structure des anneaux de Hecke d’un groupe de Chevalley fini,C. R. Acad. Sc.

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