On the unipotent support of character sheaves

TL;DR

This paper investigates Lusztig's character sheaves on reductive groups over finite fields, proving that their restriction to unipotent support is either zero or irreducible, and confirms Kawanaka's conjecture on Gelfand-Graev characters forming a basis.

Contribution

It establishes a condition under which character sheaves restrict to irreducible local systems on unipotent support and proves Gelfand-Graev characters form a basis for unipotent virtual characters.

Findings

Restriction of character sheaves is zero or irreducible on unipotent support

Gelfand-Graev characters form a Z-basis for unipotent virtual characters

Supports Kawanaka's conjecture on character bases

Abstract

Let $G$ be a connected reductive group over $F_q$, where $q$ is large enough and the center of $G$ is connected. We are concerned with Lusztig's theory of {\em character sheaves}, a geometric version of the classical character theory of the finite group $G(F_q)$. We show that under a certain technical condition, the restriction of a character sheaf to its {\em unipotent support} (as defined by Lusztig) is either zero or an irreducible local system. As an application, the generalized Gelfand-Graev characters are shown to form a $\Z$-basis of the $\Z$-module of unipotently supported virtual characters of $G(F_q)$ (Kawanaka's conjecture).