On character sheaves and characters of reductive groups at unipotent classes

TL;DR

This paper advances the understanding of character values of finite reductive groups at unipotent elements by linking character sheaves, Weyl group data, and Mellin transforms, enabling explicit computations for exceptional groups.

Contribution

It introduces new methods to compute unipotent character values using projections of Gelfand-Graev characters and Weyl group data, especially for exceptional groups.

Findings

Determined projections of Gelfand-Graev characters to unipotent characters.

Expressed character values in terms of Weyl group data.

Computed unipotent character values for exceptional groups G2, F4, E6, E7, E8.

Abstract

With a view to determining character values of finite reductive groups at unipotent elements, we prove a number of results concerning inner products of generalised Gelfand-Graev characters with characteristic functions of character sheaves, here called Lusztig functions. These are used to determine projections of generalised Gelfand-Graev characters to the space of unipo- tent characters, and to the space of characters with a given wave front set. Such projections are expressed largely in terms of Weyl group data. We show how the values of characters at their unipotent support or wave front set are determined by such data. In some exceptional groups we show that the projection of a generalised Gelfand-Graev character to a family with the same wave front set is (up to sign) the dual of a Mellin transform. Using these results, in certain cases we are able to determine roots of unity which relate almost characters to the characteristic functions. In particular we show how to compute the values of all unipotent characters at all unipotent classes for the exceptional adjoint groups of type G2, F4, E6, E7 and E8. We also pro- vide an appendix which gives a complete list of the cuspidal character sheaves on all quasi-simple groups.