Une preuve courte du principe de Selberg pour un groupe $p$-adique
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Abstract:
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).References
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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
- Email: dat@math.jussieu.fr
- Received by editor(s): June 14, 1999
- Published electronically: October 4, 2000
- Communicated by: Dan Barbasch
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 1213-1217
- MSC (2000): Primary 22E50, 22E35; Secondary 19A49
- DOI: https://doi.org/10.1090/S0002-9939-00-05834-2
- MathSciNet review: 1814155