Generic character sheaves on reductive groups over a finite ring

TL;DR

This paper constructs generic character sheaves on reductive groups over finite local rings, linking them to higher Deligne--Lusztig characters, and proves a conjecture on their perversity at level two.

Contribution

It introduces a new construction of generic character sheaves on finite local rings and proves their perversity in the level two case, extending Lusztig's results.

Findings

Construction of generic character sheaves on finite local rings

Proof of perversity conjecture at level two

Relation to higher Deligne--Lusztig characters

Abstract

In this paper we propose a construction of generic character sheaves on reductive groups over finite local rings at even levels, whose characteristic functions are higher Deligne--Lusztig characters when the parameters are generic. We formulate a conjecture on the simple perversity of these complexes, and we prove it in the level two case (thus generalised a result of Lusztig from the function field case). We then discuss the induction and restriction functors, as well as the Frobenius reciprocity, based on the perversity.