On Maximal $\mathbb{R}$-split Tori Invariant under an Involution

Abstract

Symmetric k-varieties have been a topic of interest in several fields of mathematics and physics since the 1980’s. For $k\, =\,\mathbb{R}$, symmetric $\mathbb{R}$-varieties are commonly called real symmetric spaces; however, the generalization over other fields play a role in the study of arithmetic subgroups, geometry, singularity theory, Harish Chandra modules and most importantly representation theory of Lie groups.

The preliminary study of the rationality properties of these spaces over various base fields was published by Helminck and Wang[ 1]. In order to study the representations associated with these symmetric k-varieties one needs a thorough understanding of the orbits of parabolic k-subgroups, $P_k$, acting on the symmetric k-varieties, $G_k/H_k$. This paper’s contribution is the classification of the orbits of $P\setminus G/H$ which are determined by the H-conjugacy classes of $\sigma$-stable maximal quasi k-split tori.

Mathematics Subject Classification (2010). Primary 53C35; Secondary 20C33.