Univalence of equivariant Riemann domains over the complexifications of rank-one Riemannian symmetric spaces

Laura Geatti and Andrea Iannuzzi

Vol. 238 (2008), No. 2, 275–330

DOI: 10.2140/pjm.2008.238.275

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 2000

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

Milestones

Received: 17 April 2008

Accepted: 25 July 2008

Published: 1 December 2008