Deligne-Lusztig restriction of a Gelfand-Graev module

Using Deodhar’s decomposition of a double Schubert cell, we study the regular representations of finite groups of Lie type arising in the cohomology of Deligne-Lusztig varieties associated to tori. We deduce that the Deligne-Lusztig restriction of a Gelfand-Graev module is a shifted Gelfand-Graev module. INTRODUCTION Let G be a connected reductive algebraic group defined over an algebraic closure F of a finite field of characteristic p. Let F be an isogeny of G such that some power is a Frobenius endomorphism. The finite group G = G of fixed points under F is called a finite group of Lie type. We fix a maximal torus T contained in a Borel subgroup B with unipotent radicalU, all of which assumed to be F -stable. The corresponding Weyl group will be denoted by W . In a attempt to have a complete understanding of the character theory of G , Deligne and Lusztig have introduced in [DL] a family of biadjoint morphisms Rw and Rw indexed by W , leading to an outstanding theory of induction and restriction between G and any of its maximal tori. Roughly speaking, they encode, into a virtual character, the different representations occurring in the cohomology of the corresponding Deligne-Lusztig variety. Unfortunately, the same construction does not give enough information in the modular setting, and one has to work at a higher level. More precisely, for a finite extension Λ of the ring Zl of l-adic integers, Bonnafé and Rouquier have defined in [BR1] the following functors: and Rẇ : D (ΛT -mod) −→ D(ΛG -mod) Rẇ : D (ΛG -mod) −→ D(ΛT -mod) between the derived categories of modules, which generalize the definition of Deligne-Lusztig induction and restriction. In this article we study the action of the restriction functor on a special class of representations: the Gelfand-Graev modules, which are projective modules parametrized by the G -regular characters of U . More precisely, we prove in section 3 the following result: Theorem. Let ψ : U −→ Λ be a G -regular linear character, and denote by Γψ the associated GelfandGraev module of G . Then, for any w in W , one has RẇΓψ ≃ ΛT wF [−l(w)] in the derived category D(ΛT -mod). Laboratoire de Mathématiques, Université de Franche Comté. The author is partly supported by the ANR, Project No JC07-192339.