Satake diagrams and real structures on spherical varieties

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