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Coxeter orbits and modular representations. (English) Zbl 1109.20038
G. Lusztig [Invent. Math. 38, 101-159 (1976; Zbl 0366.20031)] proved that the Frobenius eigenspaces on the \(\mathbb{Q}_l\)-cohomology spaces of the Deligne-Lusztig variety \(X\) associated to a Coxeter element are irreducible unipotent representations. In this paper the authors give a partial modular analog of this result. They study the modular representations of finite groups of Lie type arising in the cohomology of certain quotients of Deligne-Lusztig varieties associated with Coxeter elements. These quotients are related to Gelfand-Graev representations. They conjecture that the Deligne-Lusztig restriction of a Gelfand-Graev representation is a shifted Gelfand-Graev representation, and prove the conjecture for restriction to a Coxeter torus. They deduce a proof of Broué’s conjecture on equivalences of derived categories arising from Deligne-Lusztig varieties, for a split group of type \(A_n\) and a Coxeter element.

MSC:
20G05 Representation theory for linear algebraic groups
20C33 Representations of finite groups of Lie type
20G10 Cohomology theory for linear algebraic groups
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