On C-small conjugacy classes in a reductive group

TL;DR

This paper studies the existence of certain semisimple classes in reductive groups related to elliptic conjugacy classes in Weyl groups, showing they exist in good characteristic under broad conditions.

Contribution

It establishes the almost always existence of semisimple classes intersecting specific Bruhat cells in reductive groups for elliptic conjugacy classes in Weyl groups.

Findings

Semisimple classes exist in good characteristic for most elliptic conjugacy classes.

The intersection with Bruhat cells has dimension equal to that of Borel subgroups.

Results apply to almost all cases in the specified setting.

Abstract

Let G be an almost simple reductive group with Weyl group W. Let B be a Borel subgroup of G. Let C be an elliptic conjugacy class in W and let w be an element of minimal length of C. We investigate the existence of a semisimple class of G whose intersection with BwB has dimension dim(B). We show that in good characteristic such a semisimple class exists almost always.