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Une preuve courte du principe de Selberg pour un groupe $p$-adique

Author: J.-F. Dat
Journal: Proc. Amer. Math. Soc. 129 (2001), 1213-1217
MSC (2000): Primary 22E50, 22E35; Secondary 19A49
Published electronically: October 4, 2000
MathSciNet review: 1814155
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In 1992, Blanc and Brylinski showed the following property for a $p$-adic group $G$, called the ``abstract Selberg principle'': the orbital integrals on conjugacy classes of non-compact elements of the Hattori rank of a finitely generated projective smooth representation of $G$ vanish. The proof is by explicit computations of ``low'' level ($0$ and $1)$ cyclic and Hochschild cohomologies. Here we intend to show that this property is actually a direct consequence of two facts: Clozel's integration formula (which leads us to assume the defining characteristic to be zero) and the triviality of the action of unramified characters on the $K_0$ of $G$ (which is also proven here, using a standard $K$-theoretic argument due to Grothendieck).

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Additional Information

J.-F. Dat
Affiliation: Institut de Mathématiques de Jussieu, Théorie des groupes – Case 7012, 2, place Jussieu, 75251 Paris cedex 05, France

Received by editor(s): June 14, 1999
Published electronically: October 4, 2000
Communicated by: Dan Barbasch
Article copyright: © Copyright 2000 American Mathematical Society

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