Convex elements and Steinberg's cross-sections

TL;DR

This paper investigates convex elements in twisted Weyl groups, demonstrating that each conjugacy class contains such elements and that Steinberg cross-sections exist for all of them, expanding their applicability in representation theory.

Contribution

It proves the existence of convex elements in all conjugacy classes of twisted Weyl groups and establishes Steinberg cross-sections for these elements, broadening previous understanding.

Findings

Every conjugacy class contains a convex element.

Steinberg cross-sections exist for all convex elements.

Results have implications for higher Deligne-Lusztig representations.

Abstract

In this paper, we study convex elements in a (twisted) Weyl group introduced by Ivanov and the first named author. We show that each conjugacy class of the twisted Weyl group contains a convex element, and moreover, the Steinberg cross-sections exist for all convex elements. This result strictly enlarges the cases of Steinberg cross-sections from a new perspective, and will play an essential role in the study of higher Deligne-Lusztig representations.