On spherical twisted conjugacy classes

TL;DR

This paper characterizes spherical theta-twisted conjugacy classes in simple algebraic groups over algebraically closed fields of good odd characteristic, linking them to twisted involutions in the Weyl group and extending dimension formulas.

Contribution

It provides a precise characterization of spherical theta-twisted conjugacy classes and generalizes Lu's dimension formula to good odd characteristic fields.

Findings

Spherical theta-twisted conjugacy classes intersect only Bruhat cells of twisted involutions.

The characterization fails in the triality case.

Dimension formula for spherical twisted conjugacy classes is extended.

Abstract

Let G be a simple algebraic group over an algebraically closed field of good odd characteristic, and let theta be an automorphism of G arising from an involution of its Dynkin diagram. We show that the spherical theta-twisted conjugacy classes are precisely those intersecting only Bruhat cells corresponding to twisted involutions in the Weyl group. We show how the analogue of this statement fails in the triality case. We generalize to good odd characteristic J-H. Lu's dimension formula for spherical twisted conjugacy classes.