Boundary orbit strata and faces of invariant cones and complex Ol’shanskiĭ semigroups

Alldridge, Alexander

doi:10.1090/S0002-9947-2011-05309-2

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ISSN 1088-6850(online) ISSN 0002-9947(print)

Boundary orbit strata and faces of invariant cones and complex Ol'shanskiĭ semigroups

Author: Alexander Alldridge
Journal: Trans. Amer. Math. Soc. 363 (2011), 3799-3828
MSC (2010): Primary 22E60, 32M15; Secondary 22A15, 52A05
Posted: February 15, 2011
MathSciNet review: 2775828
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Abstract: Let be an irreducible Hermitian symmetric domain. Then is contained in a complexification $G_{\mathbb{C}}$ , and there exists a closed complex subsemigroup $G\subset\Gamma\subset G_{\mathbb{C}}$ , the so-called minimal Ol'shanskiĭ semigroup, characterised by the fact that all holomorphic discrete series representations of extend holomorphically to $\Gamma^\circ$ .

Parallel to the classical theory of boundary strata for the symmetric domain , due to Wolf and Korányi, we give a detailed and complete description of the -orbit type strata of $\Gamma$ as -equivariant fibre bundles. They are given by the conjugacy classes of faces of the minimal invariant cone in the Lie algebra.

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