Stein manifolds and multiplicity-free representations of compact Lie groups

TL;DR

This paper surveys recent advances in geometric representation theory, focusing on multiplicity-free representations of compact Lie groups acting on Stein manifolds, and introduces new characterizations and properties of spherical complex spaces.

Contribution

It provides new characterizations of multiplicity-free actions via antiholomorphic involutions and extends properties of spherical complex spaces to arbitrary real forms.

Findings

Characterization of actions via antiholomorphic involutions

Definition of spherical complex spaces for real forms

Equivalence of sphericity and fiber properties in homogeneous manifolds

Abstract

The paper is a survey of recent results in geometric representation theory describing group actions which induce multiplicity-free representations in the spaces of holomorphic functions. For connected compact Lie groups of automorphisms of Stein manifolds we characterize such actions in terms of antiholomorphic involutions. Some proofs are given and some results are new. For example, spherical complex spaces are defined for arbitrary real forms of complex reductive groups. Their properties, which we prove here, were known only for compact real forms. We also show that a complex compact homogeneous manifold of a complex reductive group is spherical if and only if the fiber of its Tits fibration is a complex torus.