Geometry of Hermitian symmetric spaces under the action of a maximal unipotent group

TL;DR

This paper explores the geometry of Hermitian symmetric spaces under maximal unipotent group actions, establishing conditions under which certain invariant domains are Stein and describing their holomorphic envelopes.

Contribution

It generalizes Bochner's tube theorem to Hermitian symmetric spaces, characterizing Stein domains via convexity and cone invariance of associated tube bases.

Findings

N-invariant domains are Stein iff their base is convex and cone invariant.

Provides a description of the envelope of holomorphy for N-invariant domains.

Establishes a correspondence between N-invariant domains and tube domains in Hermitian symmetric spaces.

Abstract

Let $\,G/K\,$ be a non-compact irreducible Hermitian symmetric space of rank $\,r\,$ and let $\,NAK\,$ be an Iwasawa decomposition of $\,G$. By the polydisc theorem, $\,AK/K\,$ can be regarded as the base of an $\,r$-dimensional tube domain holomorphically embedded in $\,G/K$. As every $\,N$-orbit in $\,G/K\,$ intersects $\,AK/K$ in a single point, there is a one-to-one correspondence between $\,N$-invariant domains in $\,G/K\,$ and tube domains in the product of $\,r\,$ copies of the upper half-plane in $\,\C$. In this setting we prove a generalization of Bochner's tube theorem. Namely, an $\,N$-invariant domain $\,D\,$ in $\,G/K\,$ is Stein if and only if the base $\,\Omega\,$ of the associated tube domain is convex and ``cone invariant". We also obtain a precise description of the envelope of holomorphy of an arbitrary holomorphically separable $\,N$-invariant domain over $\,G/K$.