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

 

Constant term of Eisenstein integrals on a reductive $ p$-adic symmetric space


Authors: Jacques Carmona and Patrick Delorme
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 22E50
Published electronically: April 22, 2014
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Abstract: Let $ H$ be the fixed point group of a rational involution $ \sigma $ of a reductive $ p$-adic group on a field of characteristic different from 2. Let $ P$ be a $ \sigma $-parabolic subgroup of $ G$, i.e. such that $ \sigma (P)$ is opposite $ P$. We denote the intersection $ P\cap \sigma (P)$ by $ M$.

Kato and Takano on one hand and Lagier on the other associated canonically to an $ H$-form, i.e. an $ H$-fixed linear form, $ \xi $, on a smooth admissible $ G$-module, $ V$, a linear form on the Jacquet module $ j_P(V)$ of $ V$ along $ P$ which is fixed by $ M\cap H$. We call this operation the constant term of $ H$-forms. This constant term is linked to the asymptotic behaviour of the generalized coefficients with respect to $ \xi $.

P. Blanc and the second author defined a family of $ H$-forms on certain parabolically induced representations, associated to an $ M\cap H$-form, $ \eta $, on the space of the inducing representation.

The purpose of this article is to describe the constant term of these $ H$-forms.

Also it is shown that when $ \eta $ is discrete, i.e. when the generalized coefficients of $ \eta $ are square integrable modulo the center, the corresponding family of $ H$-forms on the induced representations is a family of tempered, in a suitable sense, $ H$-forms. A formula for the constant term of Eisenstein integrals is given.


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Additional Information

Jacques Carmona
Affiliation: Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR7373, 163 Avenue de Luminy, Marseille Cedex 09, France
Email: carmona@iml.univ-mrs.fr

Patrick Delorme
Affiliation: Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR7373, 163 Avenue de Luminy, Marseille Cedex 09, France
Email: delorme@iml.univ-mrs.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-06196-5
PII: S 0002-9947(2014)06196-5
Keywords: Reductive group, nonarchimedean local field, symmetric space
Received by editor(s): December 8, 2011
Received by editor(s) in revised form: November 26, 2012
Published electronically: April 22, 2014
Additional Notes: The second author has been supported by the program ANR-BLAN08-2-326851 and by the Institut Universitaire de France during the elaboration of this work.
Article copyright: © Copyright 2014 American Mathematical Society