Spherical Stein manifolds and the Weyl involution

TL;DR

This paper characterizes spherical Stein manifolds with compact Lie group actions through the existence of a specific antiholomorphic involution that preserves orbits, linking geometric symmetry with group involutions.

Contribution

It establishes a new criterion for sphericity of Stein manifolds based on antiholomorphic involutions compatible with Weyl involutions, connecting geometric and algebraic structures.

Findings

Sphericity is equivalent to the existence of an orbit-preserving antiholomorphic involution.

Such an involution can be chosen to be equivariant with respect to a Weyl involution.

Provides a new perspective on the symmetry properties of Stein manifolds under Lie group actions.

Abstract

It is proved that a Stein manifold acted on by a connected compact Lie group is spherical if and only if there exists an antiholomorphic involution preserving each orbit of the action. This involution can be chosen equivariant with respect to a Weyl involution of the group.