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

Abstract

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

A family of adapted complexifications for SL2(R)

“…In fact, this can be regarded as a G/{±e}-equivariant, Riemann domain over G C /K C , since the subgroup {±(e, e)} of L acts trivially on m . Then, by [11,Theorem 7.6] the map q is injective and [11,Corollary 3.3] implies thatˆ m is univalent. That is,ˆ m is a Stein, L-invariant domain of G C .…”

“…For the last statement, identify P m ( m ) with m and note that its envelope of holomorphyˆ m is a Stein, L-equivariant, Riemann domain over [11,Section 3]). In fact, this can be regarded as a G/{±e}-equivariant, Riemann domain over G C /K C , since the subgroup {±(e, e)} of L acts trivially on m .…”

Invariant envelopes of holomorphy in the complexification of a Hermitian symmetric space

“…It is based on the univalence and the precise description of the envelope of holomorphy of an arbitrary Ginvariant domain in the complexification of a rank-one Hermitian symmetric space (cf. [GeIa08]). Finally, by applying Proposition 4.2(ii), one obtains D = Ξ + , The strategy is similar to the one used by Neeb in [Nee98].…”

Orbit structure of a distinguished Stein invariant domain in the complexification of a Hermitian symmetric space

“…Observe that relations (6) and (5) determine the vectors E j only up to sign. Fix an invariant complex structure J 0 of G/K .…”

“…The second goal of the paper is to prove that, in the tube-case, + contains a distinguished Stein, G-invariant subdomain S + , which arises from the compactly causal structure of a semisimple symmetric orbit G/H in the boundary of . A first evidence of this fact comes from the rank-one case SL(2, R)/S O(2, R) studied in [6], where it is also shown that every proper, Stein, invariant subdomain of + is either contained in or in S + .…”