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Geometric orbital integrals and the center of the enveloping algebra

Part of: Discontinuous groups and automorphic forms Partial differential equations on manifolds; differential operators Lie groups

Published online by Cambridge University Press:  11 August 2022

Jean-Michel Bismut  and
Shu Shen
Jean-Michel Bismut
Affiliation:
Institut de Mathématique d'Orsay, Université Paris-Saclay, Bâtiment 307, 91405 Orsay, France jean-michel.bismut@universite-paris-saclay.fr
Shu Shen
Affiliation:
Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université, Case Courrier 247, 4 place Jussieu, 75252 Paris Cedex 05, France shu.shen@imj-prg.fr
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Abstract

The purpose of this paper is to extend the explicit geometric evaluation of semisimple orbital integrals for smooth kernels for the Casimir operator obtained by the first author to the case of kernels for arbitrary elements in the center of the enveloping algebra.

Keywords

Selberg trace formulaanalysis on real and complex Lie groups

MSC classification

Primary: 11F72: Spectral theory; Selberg trace formula
Secondary: 22E30: Analysis on real and complex Lie groups 58J20: Index theory and related fixed point theorems
Type
Research Article
Information
Compositio Mathematica , Volume 158 , Issue 6 , June 2022 , pp. 1189 - 1253
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Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

The authors are much indebted to Laurent Clozel for his stimulating remarks during the preparation of the paper, and for reading the preliminary version very carefully. We thank the referee for his helpful remarks.

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