Multiplicity One and van Dijk's Conjecture for Connected Complex Symmetric Spaces

Abstract

We give new invariant-theoretic proofs of regularity for the three classical residual families DIII, balanced CII, and BDI/Spin. In DIII and balanced CII, trace contractions and a supertrace identity show that the relevant character spaces in the quadratic moment quotient vanish. These quotient-level vanishings lift to the corresponding equivariant tempered-distribution vanishings. In BDI/Spin, Gram matrices and a volume tensor give the analogous vanishing for the two single-determinant characters. We also give a direct proof of the required equivariant-distribution vanishing for EVII. These results, together with corrected distinguished-nilpotent calculations for the nice factors, establish the required componentwise regularity for the finite-component data attached to all irreducible factors and repair gaps in the previously proposed proofs. Combining these factorwise results across products and finite central quotients yields regularity for every symmetric pair of a connected complex reductive group. Generalized Harish-Chandra descent and a compatible Chevalley involution give multiplicity one for irreducible Casselman-Wallach representations in the connected-group case. A finite-component restriction argument extends this multiplicity-one statement to every possibly disconnected complex reductive $G$ whose symmetric space $G/G^θ$ is connected. Consequently, we establish van Dijk's conjecture on the unitary Gelfand property for connected complex symmetric spaces.