Satake diagrams and real structures on spherical varieties

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With each antiholomorphic involution σ of a connected complex semisimple Lie group G we associate an automorphism εσ of the Dynkin diagram. The definition of εσ is given in terms of the Satake diagram of σ. Let H ⊂ G be a self-normalizing spherical subgroup. If εσ= 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.