Invariant analytic domains in complex semisimple groups (Q677062)

Let \(H_\mathbb{R}\) be a real form of a complex, connected and simply connected, semisimple algebraic group \(H\) with complex conjugation \(\vartheta\). Let \(\Omega\) be the open dense subset of elements \(x\in H\) such that \(x\vartheta (x^{-1})\) is regular semisimple. The paper provides a study of the connected components of \(\Omega\), which are complex analytic domains, and cross-sections of the \(H_\mathbb{R} \times H_\mathbb{R}\) action on these components. This study is motivated by the Gelfand-Gindikin program to construct explicit unitary representations of \(H_\mathbb{R}\) related to complex analytic objects.