Restriction of a character sheaf to conjugacy classes

TL;DR

This paper studies how character sheaves behave when restricted to conjugacy classes in reductive groups, providing parametrizations and canonical bijections under various characteristic assumptions.

Contribution

It introduces a new understanding of the restriction of character sheaves to conjugacy classes and establishes canonical bijections relating unipotent character sheaves and Weyl group combinatorics.

Findings

Restriction of character sheaves to certain conjugacy classes yields local systems.

Parametrization of unipotent cuspidal character sheaves via conjugacy class restrictions.

Canonical bijections between unipotent character sheaves and Weyl group combinatorics.

Abstract

Let A be a character sheaf on a reductive connected group G over an algebraically closed field. Assuming that the characteristic is not bad, we show that for certain conjugacy classes D in G the restriction of A to D is a local system up to shift; we also give a parametrization of unipotent cuspidal character sheaves on G in terms of restrictions to conjugacy classes. Without restriction on characteristic we define canonical bijections from the set of unipotent character sheaves on G (and from the set of unipotent representations of the corresponding split reductive group over a finite field) to a set combinatorially defined in terms of the Weyl group.