On univalence of equivariant Riemann domains over the complexification of a non-compact, Riemannian symmetric space
Laura Geatti, Andrea Iannuzzi
Abstract
Let G/K be a non-compact, rank-one, Riemannian symmetric space and let GC be the universal complexification of G. We prove that a holomorphically separable, G-equivariant Riemann domain over GC / KC is necessarily univalent, provided that G is not a covering of SL(2, R). As a consequence of the above statement one obtains a univalence result for holomorphically separable, G x K -equivariant Riemann domains over GC. Here G x K acts on GC by left and right translations. The proof of such results involves a detailed study of the G-invariant complex geometry of the quotient GC / KC, including a complete classification of all its Stein G-invariant subdomains.
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