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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...

A real form $G$ of a complex semi-simple Lie group $G^{ℂ}$ has only finitely many orbits in any given $G^{ℂ}$-flag manifold $Z=G^{ℂ}/Q$. The complex geometry of these orbits is of interest, e.g., for the associated representation theory. The open orbits $D$ generally possess only the constant holomorphic functions, and the relevant associated geometric objects are certain positive-dimensional compact complex submanifolds of $D$ which, with very few well-understood exceptions, are parameterized by the Wolf cycle domains ${\Omega }_W\left(D\right)$ in...