ON THE COXETER COMPLEX AND ALVIS-CURTIS

DUALITY FOR PRINCIPAL ℓ-BLOCKS OF GL_{n}(q)

Markus Linckelmann and Sibylle Schroll

November 2002

Abstract. M. Cabanes and J. Rickard showed in [2] that the Alvis-Curtis character duality of a finite group of Lie type is induced in non defining characteristic ℓ by a derived equivalence given by tensoring with a bounded complex X, and they further conjecture that this derived equivalence should actually be a homotopy equivalence. Following a suggestion of R. Kessar, we show here for the special case of principal blocks of general linear groups with abelian Sylow-ℓ-subgroups that this is true, by an explicit verification relating the complex X to the Coxeter complex of the corresponding Weyl group.

Throughout this note, n is a positive integer, q a prime power and ℓ a prime divisor of q − 1 such that ℓ > n. We denote by O a complete discrete valuation ring having an algebraically closed residue field k of characteristic ℓ.

Set G = GL_{n}(q) and let b be the principal block of OG. For any ℓ′-subgroup H of G let e_{H} be the idempotent in OG defined by e_{H} = \frac{1}{|H|} ∑_{x∈H} x. Denote by T the

maximal torus of diagonal matrices in G, by U the group of upper triangular matrices whose diagonal entries are 1 and set B = UT. Let W be the subgroup of permutation matricies of G; that is W ≅ S_{n}. Denote by S the generating set of W corresponding to the set of permutations (i − 1, i), where 2 ≤ i ≤ n. For any subset I of S denote by W_{I} the subgroup of W generated by I, by P_{I} the standard parabolic subgroup of G generated by B and W_{I}, by U_{I} the unipotent radical of P_{I} and by L_{I} a Levi complement of U_{I} in P_{I}, with the convention W_{∅} = 1, U_{∅} = U, P_{∅} = B and L_{∅} = T.

Since ℓ > n, the torus T contains a Sylow-ℓ-subgroup Q of G, and then T decomposes uniquely as direct product T = Q × T′, where T′ = O_{ℓ′}(T). The set of subsets of S is viewed as simplicial complex with respect to the order which is reverse to the inclusion of subsets. The complex X defined in [2] inducing the Alvis-Curtis duality is the complex of OG-OG-bimodules associated with the coefficient system sending I ⊆ S to the OG-OG-bimodule OGe_{U_{I}} ⊗_{OP_{I}} e_{U_{I}} OG; the Coxeter complex (see e.g. [4, Section 1.15]) is

the complex C of OW -modules associated with the coefficient system sending I to the OW -module OW/W_{I}. See [1, Chapter 7] or [2] for more details on coefficient systems.

Typeset by AMS-TEX

We view X and C as cochain complexes with non zero components in the degrees 0 to |S|. The principal block b of OG has the Sylow-ℓ-subgroup Q as defect group. By a result of Puig [6, 3.4], the idempotent i = e_{T′}e_{U} is a source idempotent of b (that is, i is primitive in (OGb)^{Q} and Br_{Q}(i) ≠ 0) and there is an isomorphism of interior Q-algebras

Φ : iOGi ≅ O(Q ⋊ W) ;

that is, Φ is an O-algebra isomorphism mapping iui to u for every element u ∈ Q. The isomorphism Φ is then unique up to conjugation by an invertible element in (O(Q⋊W))^{Q}. Denote by Δ(Q ⋊ W) the diagonal subgroup of (Q ⋊ W) × (Q ⋊ W) and consider C as cochain complex of Δ(Q ⋊ W)-modules in the obvious way (that is, with ΔQ acting trivially on the components of C).

Theorem 1. With the notation above, there is an isomorphism of complexes of iOGiiOGi-bimodules

iXi ≅ Res_{Φ}(Ind_{Δ(Q⋊W)}^{(Q⋊W)×(Q⋊W)}(C)) .

In particular, iXi is homotopy equivalent to the bimodule _{σ}(iOGi) viewed as complex concentrated in degree zero, where σ is the algebra automorphism of iOGi induced via Φ by the sign representation of W.

Since OGb and iOGi are Morita equivalent via the bimodules OGi and iOG (cf. [5, 3.5]) this implies immediately the following:

Corollary 2. With the notation above, the functor X ⊗_{OG}− induces an equivalence on the

homotopy category K(OGb) of complexes of OGb-modules which extends, up to natural isomorphism, the Morita equivalence on Mod(OGb) induced by the sign representation of W.

The proof of Theorem 1 is based on the following three Propositions, the first of which is a particular case of a result of Puig [6, 3.4]:

Proposition 3. (Puig) With the notation above, for any subset I of S, the idempotent i is a source idempotent of the principal block of OP_{I} and there is an isomorphism of interior Q-algebras Φ_{I} : iOP_{I}i ≅ O(Q ⋊ W_{I}), unique up to conjugation by an invertible element in (O(Q ⋊ W_{I}))^{Q}.

Since W = W_{S} we choose notation such that Φ = Φ_{S}. The various isomorphisms Φ_{I} are compatible in the following sense:

Proposition 4. With the notation above, for any subset I of S there is an element a_{I} ∈ 1 + J((O(Q ⋊ W))^{Q}) such that Φ(iOP_{I}i) = (O(Q ⋊ W_{I}))^{a_{I}}.

Proof. For every w ∈ W_{I}, the elements w and w′ = Φ(Φ_{I}^{−1}(w)) act in the same way on Q because Φ, Φ_{I} are homomorphisms of interior Q-algebras. Thus w(w′)^{−1} ∈ ((O(Q ⋊ W))^{Q})^{×}. Since the unit element of O(Q ⋊ W) is primitive in the algebra (O(Q ⋊ W))^{Q} it follows that the group ((O(Q ⋊ W))^{Q})^{×} is isomorphic to k^{×} × (1 + J((O(Q ⋊ W))^{Q})). In other words, identifying k^{×} to its canonical preimage in O^{×} (cf. [8, Ch. II, §4, Prop. 8]) there is a unique scalar ζ(w) ∈ k^{×} such that ζ(w)^{−1}w(w′)^{−1} ∈ 1 + J((O(Q ⋊ W))^{Q}). One checks easily that the map sending w to ζ(w) is in fact a group homomorphism. Setting a_{I} = \frac{1}{|W_{I}|} ∑_{w∈W_{I}} ζ(w)^{−1}w(w′)^{−1} it is clear that a_{I} ∈ 1 + J((O(Q ⋊ W))^{Q}) and an

easy computation shows that wa_{I} = ζ(w)a_{I}w′ for any w ∈ W_{I}, implying the result. □

Proposition 5. Let I be a subset of S and denote by b_{I} the principal block of OP_{I}. We have e_{U_{I}} b = b_{I}.

Proof. Let c_{I} be the principal block of OL_{I}. Since ℓ divides q − 1 the principal blocks b and c_{I} of OG and OL_{I} are the unique unipotent blocks of OG and OL_{I}, respectively. Let χ be an irreducible character of G and let ψ be an irreducible character of L_{I} such that χ is a constituent of the Harish-Chandra induced character R_{L_{I}}^{G} (ψ). Since Harish-Chandra induction is given by tensoring with the OG-OL_{I}-bimodule OGe_{U_{I}}, this is equivalent to e_{U_{I}} e(χ)e(ψ) ≠ 0 in KG, where K is the quotient field of O and where e(χ), e(ψ) are the primitive idempotents in Z(KG), Z(KL_{I}) associated with χ, ψ, respectively. Since Harish-Chandra induction preserves Lusztig series, χ belongs to the principal block b of OG if and only if ψ belongs to the principal block c_{I} of OL_{I}. This implies e_{U_{I}} bc = 0 for any non principal block c of OL_{I}, and hence the equality e_{U_{I}} b = e_{U_{I}} bc_{I}. It implies also that e_{U_{I}} b′c_{I} = 0 for any non principal block b′ of OG, and hence the equality e_{U_{I}} bc_{I} = e_{U_{I}} c_{I}. Now U_{I} = O_{ℓ′}(P_{I}) is the kernel of the principal block of OP_{I}, from which we get the equality b_{I} = e_{U_{I}} c_{I}. The result follows. □

Proof of Theorem 1. Set Y = Ind_{Δ(Q⋊W)}^{(Q⋊W)×(Q⋊W)}(C), viewed as cochain complex of O(Q⋊ W)-O(Q ⋊ W)-bimodules. For any integer r, the degree r term of Y is isomorphic to the direct sum of the bimodules

O(Q ⋊ W) ⊗_{O(Q⋊W_{I})} O(Q ⋊ W)

with I running over the set of subsets of S such that |I| = r. The differential of Y is an alternating sum of the canonical maps

a_{IJ} : O(Q ⋊ W) ⊗_{O(Q⋊W_{I})} O(Q ⋊ W) ⟶ O(Q ⋊ W) ⊗_{O(Q⋊W_{J})} O(Q ⋊ W)

for any I ⊆ J ⊆ S such that |I| + 1 = |J|.

Let I be a subset of S. By Proposition 5, the iOGi-iOGi-bimodule iOGe_{U_{I}} ⊗_{OP_{I}}

e_{U_{I}} OGi is isomorphic to iOGe_{U_{I}} ⊗_{OP_{I}b_{I}} e_{U_{I}} OGi. Since i is still a source idempotent of

b_{I} it follows from [5, 3.5] that this bimodule is isomorphic to iOGi ⊗_{iOP_{I}i} iOGi.

Set Y′ = Res_{Φ^{−1}}(iXi). It follows from combining the Propositions 3 and 4 with the previous paragraph that the degree r term of Y′ is isomorphic to the direct sum of bimodules

O(Q ⋊ W) ⊗_{O(Q⋊W_{I})^{a_{I}}} O(Q ⋊ W)

with I running again over the set of subsets of S such that |I| = r. Furthermore, if I ⊆ J ⊆ S then Proposition 4 implies that O(Q ⋊ W_{I})^{a_{I}} ⊆ O(Q ⋊ W_{J})^{a_{J}}, and so the differential of Y′ is again just an alternating sum of the canonical maps

a′_{IJ} : O(Q ⋊ W) ⊗_{O(Q⋊W_{I})^{a_{I}}} O(Q ⋊ W) ⟶ O(Q ⋊ W) ⊗_{O(Q⋊W_{J})^{a_{J}}} O(Q ⋊ W)

for any I ⊆ J ⊆ S such that |I| + 1 = |J|.

In order to prove the first isomorphism in Theorem 1 we have to prove that Y ≅ Y′ as complexes of O(Q ⋊ W)-O(Q ⋊ W)-bimodules. There is an isomorphism

O(Q ⋊ W) ⊗_{O(Q⋊W_{I})} O(Q ⋊ W) ≅ O(Q ⋊ W) ⊗_{O(Q⋊W_{I})^{a_{I}}} O(Q ⋊ W)

mapping x ⊗ y to xa_{I} ⊗ a_{I}^{−1}y for any I ⊆ S and any x, y ∈ O(Q ⋊ W). Thus the terms of Y and Y′ are isomorphic. However, these isomorphisms need not commute to the differentials. In order to show that Y and Y′ are actually isomorphic as complexes it suffices to show that they are both split and have cohomology concentrated in degree zero.

Since ℓ does not divide the order of W the complex C is split and its cohomology cs oncentrated in degree zero isomorphic to the sign representation of W (cf. [3, 66.28] or [7, §8]). As induction is exact it follows that Y is split with cohomology concentrated in degree zero isomorphic to _{τ} O(Q ⋊ W), where τ is the automorphism of O(Q ⋊ W) mapping uw to sgn(w)uw for any u ∈ Q and any w ∈ W. It remains to show that Y′ is split with cohomology concentrated in degree zero. To see that Y′ is split we explicitly define a section s_{IJ} for the above map a′_{IJ} by setting

s_{IJ}(x ⊗ y) = \frac{1}{[W_{J} : W_{I}]}∑_{v} xva_{J}^{−1}a_{I} ⊗ a_{I}^{−1}a_{J}v^{−1}y

for any I ⊆ J ⊆ S such that |I| + 1 = |J|, any x, y ∈ O(Q ⋊ W), and where v runs over a system of representatives in W_{J} of the set of cosets W_{J}/W_{I}. A straightforward verification shows that s_{IJ} is well-defined and that a′_{IJ} ∘ s_{IJ} is the identity map.

Knowing that Y′ is split, in order to see that the cohomology is concentrated in degree zero it suffices to show that the cohomology of the quotient complex Y′/J(O(Q⋊W))Y′ is concentrated in degree zero. The point is here that the elements a_{I} are in 1 + J((O(Q ⋊ W))^{Q}) which in turn is contained in 1 + J(O(Q ⋊ W)). Thus the quotients Y/J(O(Q ⋊ W))Y and Y′/J(O(Q ⋊ W))Y′ are actually isomorphic as complexes; in particular, their cohomology is both concentrated in degree zero. Theorem 1 follows. □

Remark 6. We expect that it should be possible to extend Theorem 1 to the blocks considered by Puig in [6, 3.4], using similar techniques.

Acknowledgements. The authors would like to thank Radha Kessar for many helpful suggestions.

References

1. D. J. Benson, Representations and cohomology II, Cambridge studies in advanced mathematics 31, Cambridge University Press (1991).

2. M. Cabanes, J. Rickard, Alvis-Curtis duality as an equivalence of derived categories, in: Modular representation theory of finite groups (Charlottsville, VA, 1998), de Gruyter, Berlin (2001), 157–174.

3. C. W. Curtis, I. Reiner, Methods of representation theory II, Wiley-Interscience, New York (1987).

4. J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, Cambridge (1990).

5. L. Puig, Pointed groups and construction of characters, Math. Z. 176 (1981), 265–292.

6. L. Puig, Algèbres de source de certains blocs des groupes de Chevalley, Astérisque 181-182 (1990), 221-236.

7. J. Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London Math.Soc. 72 (1996), 331–358.

8. J.-P. Serre, Corps locaux, Hermann (Paris) (1968).

Markus Linckelmann Ohio State University, Math Tower 231 West 18th Avenue Columbus, Ohio 43210

U.S.A.

Sibylle Schroll University of North Texas P.O. Box 311430 Denton, Texas 7620-1430

U.S.A.

Université Paris 13 99, Avenue Jean-Baptiste Clément 93430 Villetaneuse

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