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Good orbital integrals


Authors: Clifton Cunningham and Thomas C. Hales
Journal: Represent. Theory 8 (2004), 414-457
MSC (2000): Primary 22E50, 14F42
DOI: https://doi.org/10.1090/S1088-4165-04-00220-1
Published electronically: September 9, 2004
MathSciNet review: 2084489
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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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  • 1. J. Adler, Refined anisotropic $K$-types and supercuspidal representations, Pacific J. Math., 185, No. 1 (1998), pp. 1-32. MR 1653184 (2000f:22019)
  • 2. J. Adler and S. DeBacker, Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive $p$-adic group, Michigan Journal of Mathematics, 2002. MR 914065 (2003g:22016)
  • 3. J. Adler and A. Roche, An intertwining result for $p$-adic groups, Canad. J. Math. 52 (2000), no. 3, 449-467. MR 1758228 (2001m:22032)
  • 4. S. DeBacker, Homogeneity results for invariant distributions of a reductive $p$-adic group. Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 3, 391-422. MR 1914003 (2003i:22019)
  • 5. François Bruhat and Jacques Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d'une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. No. 60 (1984), 197-376. MR 0756316 (86c:20042)
  • 6. R. Cluckers and F. Loeser, Fonctions constructibles et intégration motivique I, in preparation.
  • 7. J. Denef, On the degree of Igusa's local zeta function, Amer. J. Math. 109 (1987), 991-1008. MR 0919001 (89d:11108)
  • 8. J. Denef and F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Inventiones Mathematicae 135, 201-232 (1999). MR 1664700 (99k:14002)
  • 9. J. Denef, F. Loeser, Definable sets, motives, and $p$-adic integrals, Journal of the AMS, 14, No. 2, 429-469 (2001). MR 1815218 (2002k:14033)
  • 10. J. Denef, F. Loeser, Motivic Integration and the Grothendieck Group of Pseudo-Finite Fields, Proc. of the Int. Congress of Math. 2002. Vol II, 13-23, Higher Ed. Press, Beijing. MR 1957016 (2004f:14040)
  • 11. J. Denef, F. Loeser, One some rational generating series occuring in arithmetic geometry, preprint math.NT/0212202.
  • 12. H. B. Enderton, A mathematical introduction to logic. Second edition. Harcourt/ Academic Press, Burlington, MA, 2001. MR 1801397 (2001h:03001)
  • 13. M. D. Fried, M. Jarden, Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 11. Springer-Verlag, Berlin, 1986. MR 0868860 (89b:12010)
  • 14. R. Goodman, N. R. Wallach, Representations and Invariants of the Classical Groups, Encyclopedia of mathematics and its applications, v. 68, Cambridge University Press, 1998. MR 1606831 (99b:20073)
  • 15. J. Gordon and T. Hales, Virtual Transfer Factors, Represent. Theory, 7 (2003), 81-100. MR 1973368 (2004c:11084)
  • 16. M. Goresky, R. Kottwitz, and R. MacPherson, Purity of equivalued affine Springer fibers, math.RT/0305141.
  • 17. M. Goresky, R. Kottwitz, and R. MacPherson, Homology of affine Springer fibers in the unramified case, Duke Math. J., 121 (2004), no. 3, 509-561. MR 2040285
  • 18. T. C. Hales, A Simple Definition of Transfer Factors for Unramified Groups, Contemporary Math., 145, (1993) 109-134. MR 1216184 (94e:22020)
  • 19. T. C. Hales, Can $p$-adic integrals be computed?, Contributions to automorphic forms, geometry, and number theory, 313-329, Johns Hopkins Univ. Press, Baltimore, MD, 2004. MR 2058612
  • 20. T. C. Hales, Orbital Integrals are Motivic, math.RT/0212236.
  • 21. Harish-Chandra, with notes by G. van Dijk, Harmonic analysis on reductive $p$-adic groups, Lecture Notes in Mathematics 162, Springer 1970. MR 0414797 (54:2889)
  • 22. J. Kim and F. Murnaghan, Character expansions and unrefined minimal K-types, Amer. J. Math. 125 (2003), 1199-1234. MR 2018660
  • 23. J. Kim and F. Murnaghan, K-types and $\Gamma$-asymptotic expansions, preprint.
  • 24. Erasmus Landvogt, Some functorial properties of the Bruhat-Tits building, J. Reine Angew. Math. 518 (2000), pp. 213-241. MR 1739403 (2001g:20029)
  • 25. R. P. Langlands and D. Shelstad, On the definition of transfer factors, Math. Ann. 278 (1987), 219-271. MR 0909227 (89c:11172)
  • 26. G. Laumon, Sur le lemme fondamental pour les groupes unitaires, math.AG/0212245
  • 27. A. Moy and G. Prasad, Unrefined minimal K-types for $p$-adic groups, Invent. math. 116 (1994), pp. 393-408. MR 1253198 (95f:22023)
  • 28. J. Oesterlé, Réduction modulo $p^n$ des sous-ensembles analytiques fermés de $\mathbb{Z}_p^N$, Inv. Math. 66, (1982) 325-341. MR 0656627 (83j:12014)
  • 29. J. Pas, Uniform $p$-adic cell decomposition and local zeta functions. J. Reine Angew. Math. 399 (1989), 137-172. MR 1004136 (91g:11142)
  • 30. M. Presburger, Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt, Comptes-rendus du I Congrès des Mathématiciens des Pays Slaves, Warsaw, 1929, pp 92-101, 395.
  • 31. J.-P. Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math., 54 (1981) 323-401. MR 0644559 (83k:12011)
  • 32. W. Veys, Reduction modulo $p^n$ of $p$-adic subanalytic sets, Math. Proc. Camb. Phil. Soc. (1992), 112, 483-486. MR 1177996 (93i:11142)
  • 33. J.-L. Waldspurger, Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés. Astérisque No. 269 (2001). MR 1817880 (2002h:22014)
  • 34. J.-L. Waldspurger, Endoscopie et changement de caractéristique, preprint 2/2004.
  • 35. J.-L. Waldspurger, Homogénéité de certaines distributions sur les groupes $p$-adiques. Inst. Hautes Études Sci. Publ. Math. No. 81 (1995), 25-72. MR 1361755 (98f:22023)
  • 36. J.-L. Waldspurger, Une formule des traces locale pour les algèbres de Lie $p$-adiques. J. Reine Angew. Math. 465 (1995), 41-99. MR 1344131 (96i:22039)

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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: https://doi.org/10.1090/S1088-4165-04-00220-1
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
Published electronically: 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
Article copyright: © Copyright 2004 C. Cunningham and T. C. Hales

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