Let $G=K^{ℂ}$ be a complex reductive group. We give a description both of domains $\Omega \subset G$ and plurisubharmonic functions, which are invariant by the compact group, $K$, acting on $G$ by (right) translation. This is done in terms of curvature of the associated Riemannian symmetric space $M:=G/K$. Such an invariant domain $\Omega$ with a smooth boundary is Stein if and only if the corresponding domain ${\Omega }_M\subset M$ is geodesically convex and the sectional curvature of its boundary $S:=\partial {\Omega }_M$ fulfills the condition $K^S\left(E\right)\ge K^M\left(E\right)+k(E,n)$. The term $k(E,n)$ is explicitly computable...

Tuboids are tube-like domains which have a totally real edge and look asymptotically near the edge as a local tube over a convex cone. For such domains we state an analogue of Cartan’s theorem on the holomorphic convexity of totally real domains in ${ℝ}^n\subset {ℂ}^n$.

Let X be a Riemann domain over ${ℂ}^k\times {ℂ}^{\ell }$. If X is a domain of holomorphy with respect to a family ℱ ⊂(X), then there exists a pluripolar set $P\subset {ℂ}^k$ such that every slice $X_a$ of X with a∉ P is a region of holomorphy with respect to the family ${f|}_{X_a}:f\in ℱ$.