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. 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 be- tween 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 Ω 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. An important ingredient for the above results is the characterization of several classes of N-invariant plurisubharmonic funtions on D in terms of the corresponding classes of convex functions on Ω. This also leads to an explicit Lie group theoretical description of all N-invariant potentials of the Killing metric on G/K.