On the Deligne-Lusztig involution for character sheaves

TL;DR

This paper investigates the Drinfeld-Gaitsgory functor on conjugation-equivariant D-modules for reductive groups, establishing its categorical equivalence and elucidating its structure to define the Deligne-Lusztig involution conceptually.

Contribution

It provides a categorical proof that the Drinfeld-Gaitsgory functor is an equivalence and offers a new conceptual definition of the Deligne-Lusztig involution.

Findings

The Drinfeld-Gaitsgory functor is an equivalence of categories.

The functor admits a filtration with layers related to parabolic induction and restriction.

A conceptual framework for the Deligne-Lusztig involution is established.

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].