Holomorphic functions and hulls of Matsuki-dual orbits of $SU(p,q)$ in complex Grassmannians

Abstract

Let $G_0$ be a real form of a complex semisimple group acting on a flag manifold $Z$, with maximal compact subgroup $K_0$ and its complexification $K$. Matsuki duality pairs each $G_0$-orbit $γ$ with a $K$-orbit $κ$, and $M=κ\capγ$ is a compact $K_0$-orbit. For lower-dimensional $γ$, $M$ is a compact CR manifold, and we ask which functions are holomorphic near $M$ and what its holomorphic hull is. In general, we show that $K_0$-finite holomorphic functions near $M$ are regular functions on $K$-orbits, and that $M$ carries non-constant holomorphic germs if some $K$-stable prime divisor of $Z$ misses $κ$, and, under a connectedness hypothesis, only constants otherwise. For $G_0=SU(p,q)$ acting on any Grassmannian $\mathrm{Gr}_k(\mathbb{C}^{p+q})$ we give a complete answer. Holomorphic functions near $M$ are constant unless $M$ is of holomorphic type: $k\in\{p,q\}$ and $M$ lies in the Harish-Chandra chart of the bounded symmetric domain $\mathcal{D}$. Then $M$ is the manifold of rank-$c$ partial isometries, and a holomorphic hull of $M$ is $\overline{\mathcal{D}}\cap\{\mathrm{rank}\le c\}$. This agrees with the polynomial hull of Kaup and Zaitsev except in one totally real case, where it is strictly smaller because $\det^{-1}$ is holomorphic near $U(p)$. The same dichotomy decides when global CR functions on $M$ are constant and when holomorphic sections of line bundles near $M$ are global; its exceptions differ from the Gindikin-Matsuki exceptions for cycle spaces, except at the base point of $\mathcal{D}$. This settles the descent question left open in earlier work with K. Shabbir on rank-one orbits. We also determine which $K$-orbits carry non-constant regular functions, prove constancy via chains of rational curves for small $c$, and compute Levi forms in examples.