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

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., $C_H(s)=C_H(s{)}^{\circ }⋊\check{A}$ 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].

11 pages, to appear in Vietnam J. Math