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Invariant analytic domains in complex semisimple groups. (English) Zbl 0867.22004
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.
Reviewer: G.Roos (Poitiers)

22E10 General properties and structure of complex Lie groups
22E46 Semisimple Lie groups and their representations
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