Geometry of invariant domains in complex semi-simple Lie groups

TL;DR

This paper analyzes the orbit structure and CR geometry of invariant domains in complex semi-simple Lie groups, providing explicit formulas for Levi forms and establishing q-completeness of certain domains.

Contribution

It introduces explicit formulas for Levi forms of closed orbits and characterizes the Levi cone of generic orbits, advancing understanding of invariant domains in complex Lie groups.

Findings

Explicit Levi form formulas for closed orbits

Determination of Levi cones of generic orbits

Proof of q-completeness for certain invariant domains

Abstract

We investigate the joint action of two real forms of a semi-simple complex Lie group S by left and right multiplication. After analyzing the orbit structure, we study the CR structure of closed orbits. The main results are an explicit formula of the Levi form of closed orbits and the determination of the Levi cone of generic orbits. Finally, we apply these results to prove q-completeness of certain invariant domains in S.