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

Let G/K be a non-compact irreducible Hermitian symmetric space of rank r and let NAK be an Iwasawa decomposition of G. The group N acts on G/K by biholomorphisms and the real r-dimensional submanifold A center dot eK intersects every N-orbit transversally in a single point. Moreover A center dot eK is contained in a complex r-dimensional submanifold of G/K biholomorphic to H-r, the product of r copies of the upper half-plane in C. This fact leads to a one-to-one correspondence between N-invariant domains in G/K and tube domains in H-r. In this setting we prove an analogue 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 prove the univalence of N-invariant holomorphically separable Riemann domains over G/K. This yields a precise description of the envelope of holomorphy of an arbitrary N-invariant domain in G/K. Finally, we obtain a characterization of several classes of N-invariant plurisubharmonic functions on D in terms of the corresponding classes of convex functions on Omega. As an application we present an explicit Lie group theoretical description of all N-invariant potentials of the Killing metric on G/K and of the associated moment maps.