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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)

##### MSC:
 2.2e+11 General properties and structure of complex Lie groups 2.2e+47 Semisimple Lie groups and their representations
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