A Characterization of K-Invariant Stein Domains in Symmetric Embeddings

Abstract

Let K be a connected compact Lie group acting as a group of holomorphic transformations on a Stein space Ω. In this case there exists a universal complexification Ω^ℂ that is a Stein space equipped with a holomorphic K ^ℂ -action and a K-equi variant open embedding ι:Ω↩Ω^ℂ so that, if φ: Ω → Z is any holomorphic K-equivariant mapping into a K ^ℂ-space Z, there exists Ψ : Ω^ℂ → Z so that φ = Ψ ° ι[4]. Thus, when studying the complex geometry of a K-action on a Stein space Ω, we need only study K-invariant Stein domains in a Stein K ^ℂ-space X. A natural starting point is the consideration of spaces Ω that appear as fibers of the categorical quotient Ω→Ω||K; i.e., the only K-invariant holomorphic functions on Ω are the constants O(Ω)^K ≅ ℂ. It follows that Ω^ℂ is an affine K ^ℂ-space [10].