Aspects of the conjugacy class structure of simple algebraic groups

Let G be an adjoint simple algebraic group over an algebraically closed field of characteristic p; let Φ be the root system of G, and take t∈ℕ. Lawther has proven that the dimension of the set G[t]={g∈G:gt=1} depends only on Φ and t. In particular the value is independent of the characteristic p; this was observed for t small and prime by Liebeck. Since G[t] is clearly a disjoint union of conjugacy classes the question arises as to whether a similar result holds if we replace G[t] by one of those classes. This paper provides a partial answer to that question. A special case of what we have proven is the following. Take p,q to be distinct primes and Gp and Gq to be adjoint simple algebraic groups with the same root system and over algebraically closed fields of characteristic p and q respectively. If s∈Gp has order q then there exists an element u∈Gq such that o(u)=o(s) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\dim u^{G_{q}}=\dim s^{G_{p}}$\end{document} .