Abstract

Let G ∕ K be a noncompact, rank-one, Riemannian symmetric space, and let G^C be the universal complexification of G. We prove that a holomorphically separable, G-equivariant Riemann domain over G^C ∕ K^C is necessarily univalent, provided that G is not a covering of SL(2, R). As a consequence, one obtains a univalence result for holomorphically separable, G×K-equivariant Riemann domains over G^C. Here G×K acts on G^C by left and right translations. The proof of such results involves a detailed study of the G-invariant complex geometry of the quotient G^C ∕ K^C, including a complete classification of all its Stein G-invariant subdomains.

Keywords

Riemann domain, semisimple Lie group, symmetric space

Mathematical Subject Classification

Primary: 32D26, 32Q28, 53C35, 32M05

Authors