Generalized Green functions and unipotent classes for finite reductive groups, IV

TL;DR

This paper introduces the concept of split elements in unipotent classes of reductive groups, demonstrating their existence in classical and most exceptional groups, enabling the computation of generalized Green functions.

Contribution

It defines split elements for unipotent classes and proves their existence in classical and most exceptional groups, refining previous results and facilitating Green function calculations.

Findings

Split elements exist for all classical groups.

Most exceptional groups have split elements, except possibly one class in E7.

Generalized Green functions can be computed using Lusztig's algorithm with split elements.

Abstract

In this paper, we formulate the notion of split elements of a unipotent class in a connected reductive group $G$. Generalized Green functions of $G$ can be computed by using Lusztig's algorithm, if split elements exist for any unipotent class. The existence of split elements is reduced to the case where $G$ is a simply connected, almost simple group. We show, in the case of classical groups, split elements exist, which is a refinement of previous results. In the case of exceptional groups, we show the existence of split elements, possibly except one class for $G$ of type $E_7$.