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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
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
Published electronically: February 15, 2011
MathSciNet review: 2775828
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Abstract: Let $ D=G/K$ be an irreducible Hermitian symmetric domain. Then $ G$ 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 $ G$ extend holomorphically to $ \Gamma^\circ$.

Parallel to the classical theory of boundary strata for the symmetric domain $ D$, due to Wolf and Korányi, we give a detailed and complete description of the $ K$-orbit type strata of $ \Gamma$ as $ K$-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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Additional Information

Alexander Alldridge
Affiliation: Institut für Mathematik, Universität Paderborn, Warburger Strasse 100, 33100 Paderborn, Germany
Address at time of publication: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Cologne, Germany
Email: alldridg@math.upb.de, alldridg@math.uni-koeln.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05309-2
PII: S 0002-9947(2011)05309-2
Keywords: Invariant cone, complex Lie semigroup, boundary stratum, convex face, Hermitian symmetric space of non-compact type
Received by editor(s): August 25, 2009
Received by editor(s) in revised form: February 5, 2010
Published electronically: February 15, 2011
Additional Notes: This research was partially supported by the IRTG “Geometry and Analysis of Symmetries”, funded by Deutsche Forschungsgemeinschaft (DFG), Ministère de l’Éducation Nationale (MENESR), and Deutsch-Französische Hochschule (DFH-UFA)
This paper is a completely rewritten and substantially expanded version of a part of the author’s doctoral thesis under the supervision of Harald Upmeier. The author thanks him for his support and guidance. Furthermore, the author extends his thanks to an anonymous referee whose constructive criticism helped to improve the paper.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.