Stein extensions of Riemann symmetric spaces and some generalization

TL;DR

This paper generalizes the Akhiezer-Gindikin and Iwasawa domains for symmetric subgroups in real Lie groups, providing a purely algebraic proof of their inclusion relations and their connection to cycle spaces.

Contribution

It introduces generalized notions of these domains for two associated symmetric subgroups and proves their inclusion without relying on complex analysis.

Findings

Generalized Akhiezer-Gindikin and Iwasawa domains for symmetric subgroups

Purely algebraic proof of domain inclusion

Connection between domains and cycle spaces

Abstract

It was proved by Huckleberry that the Akhiezer-Gindikin domain is included in the ``Iwasawa domain'' using complex analysis. But we can see that we need no complex analysis to prove it. In this paper, we generalize the notions of the Akhiezer-Gindikin domain and the Iwasawa domain for two associated symmetric subgroups in real Lie groups and prove the inclusion. Moreover, by the symmetry of two associated symmetric subgroups, we also give a direct proof of the known fact that the Akhiezer-Gindikin domain is included in all cycle spaces.