# American Institute of Mathematical Sciences

2014, 21: 137-152. doi: 10.3934/era.2014.21.137

## Motivic functions, integrability, and applications to harmonic analysis on $p$-adic groups

 1 Université Lille 1, Laboratoire Painlevé, CNRS - UMR 8524, Cité Scientifique, 59655 Villeneuve d'Ascq Cedex, France 2 Dept. of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2, Canada 3 School of Mathematics, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, United Kingdom

Received  September 2013 Revised  August 2014 Published  October 2014

We provide a short and self-contained overview of the techniques based on motivic integration as they are applied in harmonic analysis on $p$-adic groups; our target audience is mainly representation theorists with no background in model theory (and model theorists with an interest in recent applications of motivic integration in representation theory, though we do not provide any representation theory background). We aim to give a fairly comprehensive survey of the results in harmonic analysis that were proved by such techniques in the last ten years, with emphasis on the most recent techniques and applications from [13], [8], and [32, Appendix B].
Citation: Raf Cluckers, Julia Gordon, Immanuel Halupczok. Motivic functions, integrability, and applications to harmonic analysis on $p$-adic groups. Electronic Research Announcements, 2014, 21: 137-152. doi: 10.3934/era.2014.21.137
##### References:
 [1] J. D. Adler and A. Roche, An intertwining result for $p$-adic groups,, Canad. J. Math., 52 (2000), 449. doi: 10.4153/CJM-2000-021-8. Google Scholar [2] J. D. Adler and J. Korman, The local character expansion near a tame, semisimple element,, Amer. J. Math., 129 (2007), 381. doi: 10.1353/ajm.2007.0005. Google Scholar [3] J. D. Adler and S. DeBacker, Murnaghan-Kirillov theory for supercuspidal representations of tame general linear groups,, J. Reine Angew. Math., 575 (2004), 1. doi: 10.1515/crll.2004.080. Google Scholar [4] N. Bourbaki, Variétés différentielles et analytiques. Fascicule de résultats,, Éléments de mathématique, (1967). Google Scholar [5] R. Cluckers, Presburger sets and p-minimal fields,, J. of Symbolic Logic, 68 (2003), 153. doi: 10.2178/jsl/1045861509. Google Scholar [6] R. Cluckers, G. Comte and F. Loeser, Lipschitz continuity properties for $p$-adic semi-algebraic and subanalytic functions,, Geom. Funct. Anal., 20 (2010), 68. doi: 10.1007/s00039-010-0060-0. Google Scholar [7] R. Cluckers and F. Loeser, Constructible motivic functions and motivic integration,, Invent. Math., 173 (2008), 23. doi: 10.1007/s00222-008-0114-1. Google Scholar [8] R. Cluckers, J. Gordon and I. Halupczok, Local integrability results in harmonic analysis on reductive groups in large positive characteristic,, Ann. Sci. Ec. Norm. Sup., (). Google Scholar [9] R. Cluckers, T. Hales and F. Loeser, Transfer principle for the fundamental lemma,, , (). Google Scholar [10] R. Cluckers and F. Loeser, Constructible exponential functions, motivic fourier transform and transfer principle,, Ann. of Math. (2), 171 (2010), 1011. doi: 10.4007/annals.2010.171.1011. Google Scholar [11] R. Cluckers, C. Cunningham, J. Gordon and L. Spice, On the computability of some positive-depth characters near the identity,, Represent. Theory, 15 (2011), 531. doi: 10.1090/S1088-4165-2011-00403-9. Google Scholar [12] R. Cluckers and I. Halupczok, Approximations and Lipschitz continuity in $p$-adic semi-algebraic and subanalytic geometry,, Selecta Math. (N. S.), 18 (2012), 825. doi: 10.1007/s00029-012-0088-0. Google Scholar [13] R. Cluckers, J. Gordon and I. Halupczok, Integrability of oscillatory functions on local fields: Transfer principles,, Duke Math J., 163 (2014), 1549. doi: 10.1215/00127094-2713482. Google Scholar [14] C. Cunningham and T. C. Hales, Good orbital integrals,, Represent. Theory, 8 (2004), 414. Google Scholar [15] S. DeBacker, Homogeneity results for invariant distributions of a reductive $p$-adic group,, Ann. Sci. École Norm. Sup. (4), 35 (2002), 391. doi: 10.1016/S0012-9593(02)01094-7. Google Scholar [16] S. DeBacker, Parametrizing nilpotent orbits via Bruhat-Tits theory,, Ann. of Math. (2), 156 (2002), 295. doi: 10.2307/3597191. Google Scholar [17] J. M. Diwadkar, Nilpotent conjugacy classes in $p$-adic Lie algebras: The odd orthogonal case,, Canadian J. Math., 60 (2008), 88. doi: 10.4153/CJM-2008-004-6. Google Scholar [18] J. M. Diwadkar, Nilpotent Conjugacy Classes Of Reductive $p$-adic Lie Algebras and Definability in Pas's Language,, Ph.D. Thesis, (2006). Google Scholar [19] S. Frechette, G. Gordon and L. Robson, Shalika germs for $\mathfrak{sl}_n$ and $\mathfrak{sp}_{2n}$ are motivic,, submitted., (). Google Scholar [20] J. Gordon, Motivic nature of character values of depth-zero representations,, Int. Math. Res. Not., (2004), 1735. doi: 10.1155/S1073792804133382. Google Scholar [21] B. Gross, On the motive of a reductive group,, Invent. Math., 130 (1997), 287. doi: 10.1007/s002220050186. Google Scholar [22] T. C. Hales, Orbital integrals are motivic,, Proc. Amer. Math. Soc., 133 (2005), 1515. doi: 10.1090/S0002-9939-04-07740-8. Google Scholar [23] Harish-Chandra, Admissible Invariant Distributions on Reductive $p$-adic Groups,, University Lecture Series, (1999). Google Scholar [24] J. C. Jantzen, Nilpotent orbits in representation theory,, in Lie Theory, (2004), 1. Google Scholar [25] R. E. Kottwitz, Harmonic analysis on reductive $p$-adic groups and Lie algebras,, in Harmonic Analysis, (2005), 393. Google Scholar [26] E. Lawes, Motivic Integration and the Regular Shalika Germ,, Ph.D. Thesis, (2003). Google Scholar [27] G. J. McNinch, Nilpotent orbits over ground fields of good characteristic,, Math. Ann., 329 (2004), 49. doi: 10.1007/s00208-004-0510-9. Google Scholar [28] A. Moy and G. Prasad, Unrefined minimal $K$-types for $p$-adic groups,, Invent. Math., 116 (1994), 393. doi: 10.1007/BF01231566. Google Scholar [29] M. Nevins, On nilpotent orbits of $\text{SL}_n$ and $\text{Sp}_{2n}$ over a local non-Archimedean field,, Algebr. Represent. Theory, 14 (2011), 161. doi: 10.1007/s10468-009-9182-1. Google Scholar [30] M. Presburger, On the completeness of a certain system of arithmetic of whole numbers in which addition occurs as the only operation,, Hist. Philos. Logic, 12 (1991), 225. doi: 10.1080/014453409108837187. Google Scholar [31] L. Robson, Shalika Germs Are Motivic,, M.Sc. Essay, (2012). Google Scholar [32] S. W. Shin and N. Templier, Sato-Tate theorem for families and low-lying zeroes of automorphic $l$-functions, with appendices by R. Kottwitz and J. Gordon, R. Cluckers and I. Halupczok,, , (). Google Scholar [33] T. A. Springer, Linear Algebraic Groups,, Progress in Mathematics, (1998). Google Scholar [34] J.-L. Waldspurger, Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés,, Astérisque, (2001). Google Scholar [35] A. Weil, Adeles and Algebraic Groups,, Progress in Mathematics, (1982). Google Scholar [36] A. J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function,, J. Amer. Math. Soc., 9 (1996), 1051. doi: 10.1090/S0894-0347-96-00216-0. Google Scholar [37] Z. Yun, The fundamental lemma of Jacquet and Rallis,, with an appendix by J. Gordon, 156 (2011), 167. doi: 10.1215/00127094-2010-210. Google Scholar

show all references

##### References:
 [1] J. D. Adler and A. Roche, An intertwining result for $p$-adic groups,, Canad. J. Math., 52 (2000), 449. doi: 10.4153/CJM-2000-021-8. Google Scholar [2] J. D. Adler and J. Korman, The local character expansion near a tame, semisimple element,, Amer. J. Math., 129 (2007), 381. doi: 10.1353/ajm.2007.0005. Google Scholar [3] J. D. Adler and S. DeBacker, Murnaghan-Kirillov theory for supercuspidal representations of tame general linear groups,, J. Reine Angew. Math., 575 (2004), 1. doi: 10.1515/crll.2004.080. Google Scholar [4] N. Bourbaki, Variétés différentielles et analytiques. Fascicule de résultats,, Éléments de mathématique, (1967). Google Scholar [5] R. Cluckers, Presburger sets and p-minimal fields,, J. of Symbolic Logic, 68 (2003), 153. doi: 10.2178/jsl/1045861509. Google Scholar [6] R. Cluckers, G. Comte and F. Loeser, Lipschitz continuity properties for $p$-adic semi-algebraic and subanalytic functions,, Geom. Funct. Anal., 20 (2010), 68. doi: 10.1007/s00039-010-0060-0. Google Scholar [7] R. Cluckers and F. Loeser, Constructible motivic functions and motivic integration,, Invent. Math., 173 (2008), 23. doi: 10.1007/s00222-008-0114-1. Google Scholar [8] R. Cluckers, J. Gordon and I. Halupczok, Local integrability results in harmonic analysis on reductive groups in large positive characteristic,, Ann. Sci. Ec. Norm. Sup., (). Google Scholar [9] R. Cluckers, T. Hales and F. Loeser, Transfer principle for the fundamental lemma,, , (). Google Scholar [10] R. Cluckers and F. Loeser, Constructible exponential functions, motivic fourier transform and transfer principle,, Ann. of Math. (2), 171 (2010), 1011. doi: 10.4007/annals.2010.171.1011. Google Scholar [11] R. Cluckers, C. Cunningham, J. Gordon and L. Spice, On the computability of some positive-depth characters near the identity,, Represent. Theory, 15 (2011), 531. doi: 10.1090/S1088-4165-2011-00403-9. Google Scholar [12] R. Cluckers and I. Halupczok, Approximations and Lipschitz continuity in $p$-adic semi-algebraic and subanalytic geometry,, Selecta Math. (N. S.), 18 (2012), 825. doi: 10.1007/s00029-012-0088-0. Google Scholar [13] R. Cluckers, J. Gordon and I. Halupczok, Integrability of oscillatory functions on local fields: Transfer principles,, Duke Math J., 163 (2014), 1549. doi: 10.1215/00127094-2713482. Google Scholar [14] C. Cunningham and T. C. Hales, Good orbital integrals,, Represent. Theory, 8 (2004), 414. Google Scholar [15] S. DeBacker, Homogeneity results for invariant distributions of a reductive $p$-adic group,, Ann. Sci. École Norm. Sup. (4), 35 (2002), 391. doi: 10.1016/S0012-9593(02)01094-7. Google Scholar [16] S. DeBacker, Parametrizing nilpotent orbits via Bruhat-Tits theory,, Ann. of Math. (2), 156 (2002), 295. doi: 10.2307/3597191. Google Scholar [17] J. M. Diwadkar, Nilpotent conjugacy classes in $p$-adic Lie algebras: The odd orthogonal case,, Canadian J. Math., 60 (2008), 88. doi: 10.4153/CJM-2008-004-6. Google Scholar [18] J. M. Diwadkar, Nilpotent Conjugacy Classes Of Reductive $p$-adic Lie Algebras and Definability in Pas's Language,, Ph.D. Thesis, (2006). Google Scholar [19] S. Frechette, G. Gordon and L. Robson, Shalika germs for $\mathfrak{sl}_n$ and $\mathfrak{sp}_{2n}$ are motivic,, submitted., (). Google Scholar [20] J. Gordon, Motivic nature of character values of depth-zero representations,, Int. Math. Res. Not., (2004), 1735. doi: 10.1155/S1073792804133382. Google Scholar [21] B. Gross, On the motive of a reductive group,, Invent. Math., 130 (1997), 287. doi: 10.1007/s002220050186. Google Scholar [22] T. C. Hales, Orbital integrals are motivic,, Proc. Amer. Math. Soc., 133 (2005), 1515. doi: 10.1090/S0002-9939-04-07740-8. Google Scholar [23] Harish-Chandra, Admissible Invariant Distributions on Reductive $p$-adic Groups,, University Lecture Series, (1999). Google Scholar [24] J. C. Jantzen, Nilpotent orbits in representation theory,, in Lie Theory, (2004), 1. Google Scholar [25] R. E. Kottwitz, Harmonic analysis on reductive $p$-adic groups and Lie algebras,, in Harmonic Analysis, (2005), 393. Google Scholar [26] E. Lawes, Motivic Integration and the Regular Shalika Germ,, Ph.D. Thesis, (2003). Google Scholar [27] G. J. McNinch, Nilpotent orbits over ground fields of good characteristic,, Math. Ann., 329 (2004), 49. doi: 10.1007/s00208-004-0510-9. Google Scholar [28] A. Moy and G. Prasad, Unrefined minimal $K$-types for $p$-adic groups,, Invent. Math., 116 (1994), 393. doi: 10.1007/BF01231566. Google Scholar [29] M. Nevins, On nilpotent orbits of $\text{SL}_n$ and $\text{Sp}_{2n}$ over a local non-Archimedean field,, Algebr. Represent. Theory, 14 (2011), 161. doi: 10.1007/s10468-009-9182-1. Google Scholar [30] M. Presburger, On the completeness of a certain system of arithmetic of whole numbers in which addition occurs as the only operation,, Hist. Philos. Logic, 12 (1991), 225. doi: 10.1080/014453409108837187. Google Scholar [31] L. Robson, Shalika Germs Are Motivic,, M.Sc. Essay, (2012). Google Scholar [32] S. W. Shin and N. Templier, Sato-Tate theorem for families and low-lying zeroes of automorphic $l$-functions, with appendices by R. Kottwitz and J. Gordon, R. Cluckers and I. Halupczok,, , (). Google Scholar [33] T. A. Springer, Linear Algebraic Groups,, Progress in Mathematics, (1998). Google Scholar [34] J.-L. Waldspurger, Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés,, Astérisque, (2001). Google Scholar [35] A. Weil, Adeles and Algebraic Groups,, Progress in Mathematics, (1982). Google Scholar [36] A. J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function,, J. Amer. Math. Soc., 9 (1996), 1051. doi: 10.1090/S0894-0347-96-00216-0. Google Scholar [37] Z. Yun, The fundamental lemma of Jacquet and Rallis,, with an appendix by J. Gordon, 156 (2011), 167. doi: 10.1215/00127094-2010-210. Google Scholar
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