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

Let $G\mathrm{/}K$ be a non-compact irreducible Hermitian symmetric space of rank $r$ and let $NAK$ be an Iwasawa decomposition of $G$. By the polydisc theorem, $AK\mathrm{/}K$ can be regarded as the base of an $r$-dimensional tube domain holomorphically embedded in $G\mathrm{/}K$. As every $N$-orbit in $G\mathrm{/}K$ intersects $AK\mathrm{/}K$ in a single point, there is a one-to-one correspondence between $N$-invariant domains in $G\mathrm{/}K$ and tube domains in the product of $r$ copies of the upper half-plane in $\text{C}$. In this setting we prove a generalization of Bochner’s tube theorem. Namely, an $N$-invariant domain $D$ in $G\mathrm{/}K$ is Stein if and only if the base $\mathrm{\Omega }$ 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 $G\mathrm{/}K$.