On semisimple classes and component groups in type $\mathsf{D}$

TL;DR

This paper investigates the structure of centralizers of semisimple elements in type D algebraic groups, providing new insights into automorphism actions, character parametrization, and contributing to the proof of the McKay conjecture for prime 3.

Contribution

It demonstrates that centralizers of semisimple elements split over their connected components in type D groups, aiding in character theory and automorphism analysis.

Findings

Centralizers split over their connected components in type D groups.

Automorphisms act predictably on semisimple conjugacy classes.

Results contribute to the proof of the McKay conjecture for prime 3.

Abstract

In adjoint reductive groups $H$ of type $\mathsf{D}$ we show that for every semisimple element $s$, its centralizer splits over its connected component, i.e., $C_H(s) = C_H(s)^\circ \rtimes \check A$ for some complement $\check A$ with strong stability properties. We derive several consequences about the action of automorphisms on semisimple conjugacy classes. This helps to parametrize characters of the finite groups $\mathsf{D}_{l,\text{sc}}(q)$ and $^2\mathsf{D}_{l,\text{sc}}(q)$ and describe the action of automorphisms on them. It is also a contribution to the final proof of the McKay conjecture for the prime 3, see [S21], [S23].