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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

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A class of parabolic $k$-subgroups associated with symmetric $k$-varieties
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by A. G. Helminck and G. F. Helminck PDF
Trans. Amer. Math. Soc. 350 (1998), 4669-4691 Request permission

Abstract:

Let $G$ be a connected reductive algebraic group defined over a field $k$ of characteristic not 2, $\sigma$ an involution of $G$ defined over $k$, $H$ a $k$-open subgroup of the fixed point group of $\sigma$, $G_k$ (resp. $H_k$) the set of $k$-rational points of $G$ (resp. $H$) and $G_k/H_k$ the corresponding symmetric $k$-variety. A representation induced from a parabolic $k$-subgroup of $G$ generically contributes to the Plancherel decomposition of $L^2(G_k/H_k)$ if and only if the parabolic $k$-subgroup is $\sigma$-split. So for a study of these induced representations a detailed description of the $H_k$-conjucagy classes of these $\sigma$-split parabolic $k$-subgroups is needed. In this paper we give a description of these conjugacy classes for general symmetric $k$-varieties. This description can be refined to give a more detailed description in a number of cases. These results are of importance for studying representations for real and $\mathfrak p$-adic symmetric $k$-varieties.
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Additional Information
  • A. G. Helminck
  • Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina, 27695-8205
  • Email: loek@math.ncsu.edu
  • G. F. Helminck
  • Affiliation: Department of Mathematics, Universiteit Twente, Enschede, The Netherlands
  • Email: helminck@math.utwente.nl
  • Received by editor(s): December 15, 1995
  • Received by editor(s) in revised form: December 15, 1996
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 4669-4691
  • MSC (1991): Primary 20G15, 20G20, 22E15, 22E46, 53C35
  • DOI: https://doi.org/10.1090/S0002-9947-98-02029-7
  • MathSciNet review: 1443876