Classification of spherical nilpotent orbits for $U(p,p)$

Abstract

We consider the symmetric pair $(G,K) = (U(p, q), U(p) \times U(q))$. For this pair, we classify spherical nilpotent $K_{\mathbb{C}}$-orbits which are theta lift in the stable range. For the pair $(G,K) = (U(p, p), U(p) \times U(p))$ where $p = q$, we prove that a spherical nilpotent $K_{\mathbb{C}}$-orbit must be a theta lift. As a consequence, we get a complete classification of the spherical nilpotent $K_{\mathbb{C}}$-orbits for the symmetric pair $(U(p, p), U(p) \times U(p))$.