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Good orbital integrals

Author(s): Clifton Cunningham; Thomas C. Hales
Journal: Represent. Theory 8 (2004), 414-457.
MSC (2000): Primary 22E50, 14F42
Posted: September 9, 2004
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Abstract: This paper concerns a class of orbital integrals in Lie algebras over $p$-adic fields. The values of these orbital integrals at the unit element in the Hecke algebra count points on varieties over finite fields. The construction, which is based on motivic integration, works both in characteristic zero and in positive characteristic. As an application, the Fundamental Lemma for this class of integrals is lifted from positive characteristic to characteristic zero. The results are based on a formula for orbital integrals as distributions inflated from orbits in the quotient spaces of the Moy-Prasad filtrations of the Lie algebra. This formula is established by Fourier analysis on these quotient spaces.

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Additional Information:

Clifton Cunningham
Affiliation: Department of Mathematics, University of Calgary, Alberta, Canada, T2N 1N4
Email: cunning@math.ucalgary.ca

Thomas C. Hales
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260
Email: hales@pitt.edu

DOI: 10.1090/S1088-4165-04-00220-1
PII: S 1088-4165(04)00220-1
Keywords: Orbital integrals, local constancy, motivic integration, Fundamental Lemma
Received by editor(s): November 21, 2003
Received by editor(s) in revised form: April 27, 2004
Posted: September 9, 2004
Additional Notes: The research of the second author was supported in part by the NSF
This work is licensed under the Creative Commons Attribution License. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA
Copyright of article: Copyright 2004, C. Cunningham and T. C. Hales

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