$\theta$-semisimple twisted conjugacy classes of type D in ${\mathrm{PSL}}_n(q)$

Abstract

Let $p$ be an odd prime, $m\in \mathbb{N}$ and set $q=p^m$, $G={\mathrm{PSL}}_n(q)$. Let $\theta$ be a standard graph automorphism of $G$, $d$ be a diagonal automorphism and ${\mathrm{Fr}}_q$ be the Frobenius endomorphism of ${\mathrm{PSL}}_n(\overline{{\mathbb{F}}_q})$. We show that every $(d\circ \theta )$-conjugacy class of a $(d\circ \theta ,p)$-regular element in $G$ is represented in some ${\mathrm{Fr}}_q$-stable maximal torus and that most of them are of type D. We write out the possible exceptions and show that, in particular, if $n\ge 5$ is either odd or a multiple of $4$ and $q>7$, then all such classes are of type D. We develop general arguments to deal with twisted classes in finite groups.

24 pages. Accepted for publication in Journal of Lie Theory