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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A class of parabolic $k$-subgroups
associated with symmetric $k$-varieties


Authors: A. G. Helminck and G. F. Helminck
Journal: Trans. Amer. Math. Soc. 350 (1998), 4669-4691
MSC (1991): Primary 20G15, 20G20, 22E15, 22E46, 53C35
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02029-7
PII: S 0002-9947(98)02029-7
Received by editor(s): December 15, 1995
Received by editor(s) in revised form: December 15, 1996
Article copyright: © Copyright 1998 American Mathematical Society