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Let X be a holomorphically separable irreducible reduced complex space, K a connected compact Lie group acting on X by holomorphic transformations, θ: K → K a Weyl involution, and μ: X → X an antiholomorphic map satisfying μ 2 = Id and μ(kx) = θ(k)μ(x) for x ∈ X, k ∈ K. We show that if 0(X) is a multiplicity free K-module, then μ maps every K-orbit onto itself. For a spherical affine homogeneous space X = G/H of the reductive group G = K C we construct an antiholomorphic map μ with these properties.

Antiholomorphic involutions of spherical complex spaces