Correspondence of Spherical Distributions Between Symmetrical Spaces of Type GC/GR

Let G be a complex, connected and simply connected semisimple Lie group with Lie algebra g. Let σ and τ be two involutions of g such that (1) the algebras gσ and gτ of points fixed by σ and τ, respectively, are real forms of g, (2) the group Gτ of points fixed by τ is quasisplit, and (3) the automorphism σ τ is inner. We characterize the stable invariant integral of a C∞ function with compact support on G/Gσ. Then, we generalize the results of Harinck to the case of a reductive symmetric space of type GC/GR. This allows us to construct stable spherical distributions on G/Gσ. Afterward, we construct a correspondence between stable spherical distributions on G/Gτ and those of G/Gσ.