On univalence of equivariant Riemann domains over the complexification of a non-compact, Riemannian symmetric space

摘要

Let G/K be a non-compact, rank-one, Riemannian symmetric space and let G^C bethe universal complexification of G. We prove that a holomorphically separable,G-equivariant Riemann domain over G^C / K^C is necessarily univalent, providedthat G is not a covering of SL(2, R). As a consequence of the above statementone obtains a univalence result for holomorphically separable, G x K-equivariant Riemann domains over G^C. Here G x K acts on G^C by left and righttranslations. The proof of such results involves a detailed study of theG-invariant complex geometry of the quotient G^C / K^C, including a completeclassification of all its Stein G-invariant subdomains.