A Characterization of K-Invariant Stein Domains in Symmetric Embeddings

  • Alan T. Huckleberry
  • G. Fels
Part of the The University Series in Mathematics book series (USMA)


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 that is a Stein space equipped with a holomorphic K -action and a K-equi variant open embedding ι:Ω↩Ω so that, if φ: Ω → Z is any holomorphic K-equivariant mapping into a K -space Z, there exists Ψ : Ω → 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 -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 is an affine K -space [10].


Weyl Group Open Orbit Categorical Quotient Reinhardt Domain Stein Space 
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Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Alan T. Huckleberry
    • 1
  • G. Fels
    • 1
  1. 1.Institut für MathematikRuhr-Universität BochumBochum 1Germany

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