Polynomially convex orbits of compact lie groups

Abstract

LetV be a finite dimensional complex linear space and letG be a compact subgroup of GL(V). We prove that an orbitGυ, υ ∈ V, is polynomially convex if and only ifG ^ℂυ is closed andGυ is the real form ofG ^ℂυ. For every orbitGυ which is not polynomially convex we construct an analytic annulus or strip inG ^ℂυ with the boundary inGυ. It is also proved that the group of holomorphic automorphisms ofG ^ℂυ which commute withG ^ℂ acts transitively on the set of polynomially convexG-orbits. Further, an analog of the Kempf-Ness criterion is obtained and homogeneous spaces of compact Lie groups which admit only polynomially convex equivariant embeddings are characterized.