On a Dimension Formula for Twisted Spherical Conjugacy Classes in Semisimple Algebraic Groups

Let $G$ be a connected semisimple algebraic group over an algebraically closed field of characteristic zero, and let $þ$ be an automorphism of $G$. We give a characterization of $þ$-twisted spherical conjugacy classes in $G$ by a formula for their dimensions in terms of certain elements in the Weyl group of $G$, generalizing a result of N. Cantarini, G. Carnovale, and M. Costantini when $þ$ is the identity automorphism. For $G$ simple and $þ$ an outer automorphism of $G$, we also classify the Weyl group elements that appear in the dimension formula.

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