The local structure theorem for real spherical varieties

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 $Q \times_L S$. Here $Q$ is a parabolic subgroup with Levi decomposition $LU$, and $S$ is a homogeneous space for a quotient $D=L/L_n$ of $L$, where $L_n$ is normal in $L$, such that $D$ is compact modulo center.