Satake diagrams and real structures on spherical varieties

Abstract

With each antiholomorphic involution $σ$ of a connected complex semisimple Lie group $G$ we associate an automorphism $ε_σ$ of the Dynkin diagram. The definition of $ε_σ$ is given in terms of the Satake diagram of $σ$. Let $H \subset G$ be a self-normalizing spherical subgroup. If $ε_σ={\rm id}$ then we prove the uniqueness and existence of a $σ$-equivariant real structure on $G/H$ and on the wonderful completion of $G/H$.