Function Theory on Homogeneous Manifolds

Abstract

Let K be a connected compact Lie group, G = K_c the reductive linear algebraic group obtained by complexification, and H ⊂ G a closed complex Lie subgroup. In this chapter we study holomorphic functions in K-invariant domains Ω ⊂ G/H. For any such domain there is a representation of K on the Fréchet vector space O(Ω). Therefore our starting point is a theorem of Harish-Chandra, which extends the classical Fourier expansion to the representation theory of compact Lie groups on Fréchet spaces. As an application, we prove that for G/H holomorphically separable the subgroup H is closed in the Zariski topology of G. Furthermore, under this assumption G/H is a quasi-affine algebraic variety. Algebraic subgroups of (not necessarily reductive) linear algebraic groups having this property are called observable. An algebraic subgroup H ⊂ G is observable if and only if G/H is an orbit in a finite-dimensional rational G-module. Using the methods of the geometric invariant theory, we obtain a description of the class of observable subgroups. Namely, an algebraic subgroup H of a connected linear algebraic group G is observable if and only if there exist an irreducible rational G-module V and a vector υ ∈ V with G[υ] closed in P(V), such that H ⊂ G _υ and the unipotent radical of H is contained in the unipotent radical of G _υ . If G is reductive, then this algebraic condition is necessary and sufficient for G/H to be holomorphically separable.