Regularity and the Gelfand Property for Complex Symmetric Pairs

Abstract

We prove that every symmetric pair of a connected complex reductive group is regular in the sense of Aizenbud--Gourevitch. This settles the Aizenbud--Gourevitch regularity conjecture over the complex numbers. Generalized Harish--Chandra descent then makes the canonical central cover of every complex symmetric pair a Gelfand--Kazhdan pair. An anti-automorphism arising from a compatible Chevalley involution upgrades the resulting GP2 bound to GP1 on the cover, and finite central descent transfers GP1 to the original pair. In particular, van Dijk's conjecture on complex symmetric pairs follows. Rubio reduced the unresolved irreducible regularity problem to four families: the DIII family $(D_r,A_{r-1}+\mathbb{C})$, the balanced CII family $(C_{2r},C_r+C_r)$, some remaining Spin block pairs, and the EVII pair $(E_7,E_6+\mathbb{C})$. We treat these cases by four different mechanisms. For DIII we construct a sign-equivariant Schwartz distribution on the regular set and extend it across a common orbit boundary by the Chen--Sun theorem. For balanced CII we combine homogeneity, distinguished nilpotent orbits, and a stable-density theorem for the centralizer representation. For Spin blocks we prove pleasantness for unequal odd--odd blocks, use Przebinda's orthogonal-distribution theorem in odd smaller rank, and construct a finite orbit closure with automatic extension in even smaller rank. For EVII we compute the graded-$\mathfrak{sl}_2$ data for all twenty-two nilpotent orbits and use central-torus characters to eliminate the remaining resonances, including the two residual triple-centralizer cases. A finite-component assembly theorem then handles arbitrary connected central quotients and diagonal couplings among simple factors.