The local structure theorem for real spherical varieties. (9th July 2015) - Knop, Friedrich; Krötz, Bernhard; Schlichtkrull, Henrik
Abstract : Let $G$ be an algebraic real reductive group and $Z$ a real spherical $G$ -variety, that is, it admits an open orbit for a minimal parabolic subgroup $P$ . We prove a local structure theorem for $Z$ . In the simplest case where $Z$ is homogeneous, the theorem provides an isomorphism of the open $P$ -orbit with a bundle $Q× _{L}S$ . Here $Q$ is a parabolic subgroup with Levi decomposition $L\ltimes U$, and $S$ is a homogeneous space for a quotient $D=L/L_{n}$ of $L$, where $L_{n}\subseteq L$ is normal, such that $D$ is compact modulo center.