Decompositions related to symmetric varieties

Let G be a connected reductive group over an algebraically closed field of characteristic ≠2 and let σ be an automorphism of G of order 2, whose fixed point group is K. The paper establishes for certain G-varieties X a decomposition of X in finitely many K-stable locally closed pieces. Namely, (a) if X=G/P, where P is a parabolic subgroup of G and (b) (G adjoint) if X is the wonderful compactification of the symmetric variety G/K.