On the Deligne-Lusztig involution for character sheaves

Abstract

For a reductive group G, we study the Drinfeld-Gaitsgory functor of the category of conjugation-equivariant D-modules on G. We show that this functor is an equivalence of categories, and that it has a filtration with layers expressed via parabolic induction of parabolic restriction. We use this to provide a conceptual definition of the Deligne-Lusztig involution on the set of isomorphism classes of irreducible character D-modules, which was defined previously in [Lu1, section 15].

Current re-upload: Fixed an error in formulation of Claim 3.1 (should require more continuity than previously stated, in order for claimed to hold), and accordingly fixed formulation of Claim 3.2, Corollary 3.3, and wordings of proof of Proposition 5.1 and proof of Proposition 5.5 (thanks to D. Gaitsgory for pointing out this error)