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Multiplicity bounds and the subrepresentation theorem for real spherical spaces

Authors: Bernhard Krötz and Henrik Schlichtkrull
Journal: Trans. Amer. Math. Soc. 368 (2016), 2749-2762
MSC (2010): Primary 22E45, 43A85
Published electronically: November 24, 2014
MathSciNet review: 3449256
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be a real semi-simple Lie group and $ H$ a closed subgroup which admits an open orbit on the flag manifold of a minimal parabolic subgroup. Let $ V$ be a Harish-Chandra module. A uniform finite bound is given for the dimension of the space of $ H$-fixed distribution vectors for $ V$, and a related subrepresentation theorem is derived.

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  • [1] Erik P. van den Ban, Invariant differential operators on a semisimple symmetric space and finite multiplicities in a Plancherel formula, Ark. Mat. 25 (1987), no. 2, 175-187. MR 923405 (89g:22019),
  • [2] Erik P. van den Ban, Asymptotic behaviour of matrix coefficients related to reductive symmetric spaces, Nederl. Akad. Wetensch. Indag. Math. 49 (1987), no. 3, 225-249. MR 914083 (89c:22025)
  • [3] Erik P. van den Ban, The principal series for a reductive symmetric space. I. $ H$-fixed distribution vectors, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 3, 359-412. MR 974410 (90a:22016)
  • [4] William Casselman, Jacquet modules for real reductive groups, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 557-563. MR 562655 (83h:22025)
  • [5] William Casselman and Dragan Miličić, Asymptotic behavior of matrix coefficients of admissible representations, Duke Math. J. 49 (1982), no. 4, 869-930. MR 683007 (85a:22024),
  • [6] Patrick Delorme, Injection de modules sphériques pour les espaces symétriques réductifs dans certaines représentations induites, with an appendix by Erik van den Ban and Delorme, Noncommutative harmonic analysis and Lie groups (Marseille-Luminy, 1985), Lecture Notes in Math., vol. 1243, Springer, Berlin, 1987, pp. 108-143 (French). MR 897540 (89c:22026),
  • [7] Henryk Hecht and Wilfried Schmid, Characters, asymptotics and $ {\mathfrak{n}}$-homology of Harish-Chandra modules, Acta Math. 151 (1983), no. 1-2, 49-151. MR 716371 (84k:22026),
  • [8] Anthony W. Knapp, Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR 855239 (87j:22022)
  • [9] Anthony W. Knapp, Lie groups beyond an introduction, 2nd ed., Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1920389 (2003c:22001)
  • [10] Toshiyuki Kobayashi and Toshio Oshima, Finite multiplicity theorems for induction and restriction, Adv. Math. 248 (2013), 921-944. MR 3107532,
  • [11] Friedrich Knop, Bernhard Krötz, and Henrik Schlichtkrull, The local structure theorem for real spherical spaces, submitted, arXiv:1310.6390
  • [12] Bernhard Krötz and Henrik Schlichtkrull, Finite orbit decomposition of real flag manifolds, to appear in J. Eur. Math. Soc.
  • [13] Bernhard Krötz, Eitan Sayag, and Henrik Schlichtkrull, Decay of matrix coefficients on reductive homogeneous spaces of spherical type, Math. Z. 278 (2014), no. 1-2, 229-249. MR 3267577,
  • [14] Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988. MR 929683 (89i:22029)

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Additional Information

Bernhard Krötz
Affiliation: Institut für Mathematik, Universität Paderborn, Warburger Straße 100, D-33098 Paderborn, Germany

Henrik Schlichtkrull
Affiliation: Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark

Keywords: Homogeneous space, Harish-Chandra module, subrepresentation theorem
Received by editor(s): September 4, 2013
Received by editor(s) in revised form: November 22, 2013, January 14, 2014, and March 8, 2014
Published electronically: November 24, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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