$\theta$-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.

arXiv comment24 pages. Accepted for publication in Journal of Lie Theory