# 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), 449-467. doi: 10.4153/CJM-2000-021-8. [2] J. D. Adler and J. Korman, The local character expansion near a tame, semisimple element, Amer. J. Math., 129 (2007), 381-403. doi: 10.1353/ajm.2007.0005. [3] J. D. Adler and S. DeBacker, Murnaghan-Kirillov theory for supercuspidal representations of tame general linear groups, J. Reine Angew. Math., 575 (2004), 1-35. doi: 10.1515/crll.2004.080. [4] N. Bourbaki, Variétés différentielles et analytiques. Fascicule de résultats, Éléments de mathématique, Fasc. XXXIII, Actualités Scientifiques et Industrielles, Hermann, Paris, 1967. [5] R. Cluckers, Presburger sets and p-minimal fields, J. of Symbolic Logic, 68 (2003), 153-162. doi: 10.2178/jsl/1045861509. [6] R. Cluckers, G. Comte and F. Loeser, Lipschitz continuity properties for $p$-adic semi-algebraic and subanalytic functions, Geom. Funct. Anal., 20 (2010), 68-87. doi: 10.1007/s00039-010-0060-0. [7] R. Cluckers and F. Loeser, Constructible motivic functions and motivic integration, Invent. Math., 173 (2008), 23-121. doi: 10.1007/s00222-008-0114-1. [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., (). [9] R. Cluckers, T. Hales and F. Loeser, Transfer principle for the fundamental lemma,, , (). [10] R. Cluckers and F. Loeser, Constructible exponential functions, motivic fourier transform and transfer principle, Ann. of Math. (2), 171 (2010), 1011-1065. doi: 10.4007/annals.2010.171.1011. [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), 531-567. doi: 10.1090/S1088-4165-2011-00403-9. [12] R. Cluckers and I. Halupczok, Approximations and Lipschitz continuity in $p$-adic semi-algebraic and subanalytic geometry, Selecta Math. (N. S.), 18 (2012), 825-837. doi: 10.1007/s00029-012-0088-0. [13] R. Cluckers, J. Gordon and I. Halupczok, Integrability of oscillatory functions on local fields: Transfer principles, Duke Math J., 163 (2014), 1549-1600. doi: 10.1215/00127094-2713482. [14] C. Cunningham and T. C. Hales, Good orbital integrals, Represent. Theory, 8 (2004), 414-457 (electronic). [15] S. DeBacker, Homogeneity results for invariant distributions of a reductive $p$-adic group, Ann. Sci. École Norm. Sup. (4), 35 (2002), 391-422. doi: 10.1016/S0012-9593(02)01094-7. [16] S. DeBacker, Parametrizing nilpotent orbits via Bruhat-Tits theory, Ann. of Math. (2), 156 (2002), 295-332. doi: 10.2307/3597191. [17] J. M. Diwadkar, Nilpotent conjugacy classes in $p$-adic Lie algebras: The odd orthogonal case, Canadian J. Math., 60 (2008), 88-108. doi: 10.4153/CJM-2008-004-6. [18] J. M. Diwadkar, Nilpotent Conjugacy Classes Of Reductive $p$-adic Lie Algebras and Definability in Pas's Language, Ph.D. Thesis, University of Pittsburgh, 2006. [19] S. Frechette, G. Gordon and L. Robson, Shalika germs for $\mathfrak{sl}_n$ and $\mathfrak{sp}_{2n}$ are motivic,, submitted., (). [20] J. Gordon, Motivic nature of character values of depth-zero representations, Int. Math. Res. Not., (2004), 1735-1760. doi: 10.1155/S1073792804133382. [21] B. Gross, On the motive of a reductive group, Invent. Math., 130 (1997), 287-313. doi: 10.1007/s002220050186. [22] T. C. Hales, Orbital integrals are motivic, Proc. Amer. Math. Soc., 133 (2005), 1515-1525. doi: 10.1090/S0002-9939-04-07740-8. [23] Harish-Chandra, Admissible Invariant Distributions on Reductive $p$-adic Groups, University Lecture Series, 16, American Mathematical Society, 1999. [24] J. C. Jantzen, Nilpotent orbits in representation theory, in Lie Theory, Progr. Math., 288, Birkhäuser Boston, Boston, MA, 2004, 1-211. [25] R. E. Kottwitz, Harmonic analysis on reductive $p$-adic groups and Lie algebras, in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Math. Proc., 4, Amer. Math. Soc., Providence, RI, 2005, 393-522. [26] E. Lawes, Motivic Integration and the Regular Shalika Germ, Ph.D. Thesis, University of Michigan, 2003. [27] G. J. McNinch, Nilpotent orbits over ground fields of good characteristic, Math. Ann., 329 (2004), 49-85. doi: 10.1007/s00208-004-0510-9. [28] A. Moy and G. Prasad, Unrefined minimal $K$-types for $p$-adic groups, Invent. Math., 116 (1994), 393-408. doi: 10.1007/BF01231566. [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), 161-190. doi: 10.1007/s10468-009-9182-1. [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), 225-233. doi: 10.1080/014453409108837187. [31] L. Robson, Shalika Germs Are Motivic, M.Sc. Essay, University of British Columbia, 2012. [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,, , (). [33] T. A. Springer, Linear Algebraic Groups, Progress in Mathematics, 9, Birkhäuser, Boston Inc, 1998. [34] J.-L. Waldspurger, Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés, Astérisque, No. 269 (2001). [35] A. Weil, Adeles and Algebraic Groups, Progress in Mathematics, 23, Birkhäuser Basel, 1982. [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), 1051-1094. doi: 10.1090/S0894-0347-96-00216-0. [37] Z. Yun, The fundamental lemma of Jacquet and Rallis, with an appendix by J. Gordon, Duke Math. J., 156 (2011), 167-227. doi: 10.1215/00127094-2010-210.

show all references

##### References:
 [1] J. D. Adler and A. Roche, An intertwining result for $p$-adic groups, Canad. J. Math., 52 (2000), 449-467. doi: 10.4153/CJM-2000-021-8. [2] J. D. Adler and J. Korman, The local character expansion near a tame, semisimple element, Amer. J. Math., 129 (2007), 381-403. doi: 10.1353/ajm.2007.0005. [3] J. D. Adler and S. DeBacker, Murnaghan-Kirillov theory for supercuspidal representations of tame general linear groups, J. Reine Angew. Math., 575 (2004), 1-35. doi: 10.1515/crll.2004.080. [4] N. Bourbaki, Variétés différentielles et analytiques. Fascicule de résultats, Éléments de mathématique, Fasc. XXXIII, Actualités Scientifiques et Industrielles, Hermann, Paris, 1967. [5] R. Cluckers, Presburger sets and p-minimal fields, J. of Symbolic Logic, 68 (2003), 153-162. doi: 10.2178/jsl/1045861509. [6] R. Cluckers, G. Comte and F. Loeser, Lipschitz continuity properties for $p$-adic semi-algebraic and subanalytic functions, Geom. Funct. Anal., 20 (2010), 68-87. doi: 10.1007/s00039-010-0060-0. [7] R. Cluckers and F. Loeser, Constructible motivic functions and motivic integration, Invent. Math., 173 (2008), 23-121. doi: 10.1007/s00222-008-0114-1. [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., (). [9] R. Cluckers, T. Hales and F. Loeser, Transfer principle for the fundamental lemma,, , (). [10] R. Cluckers and F. Loeser, Constructible exponential functions, motivic fourier transform and transfer principle, Ann. of Math. (2), 171 (2010), 1011-1065. doi: 10.4007/annals.2010.171.1011. [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), 531-567. doi: 10.1090/S1088-4165-2011-00403-9. [12] R. Cluckers and I. Halupczok, Approximations and Lipschitz continuity in $p$-adic semi-algebraic and subanalytic geometry, Selecta Math. (N. S.), 18 (2012), 825-837. doi: 10.1007/s00029-012-0088-0. [13] R. Cluckers, J. Gordon and I. Halupczok, Integrability of oscillatory functions on local fields: Transfer principles, Duke Math J., 163 (2014), 1549-1600. doi: 10.1215/00127094-2713482. [14] C. Cunningham and T. C. Hales, Good orbital integrals, Represent. Theory, 8 (2004), 414-457 (electronic). [15] S. DeBacker, Homogeneity results for invariant distributions of a reductive $p$-adic group, Ann. Sci. École Norm. Sup. (4), 35 (2002), 391-422. doi: 10.1016/S0012-9593(02)01094-7. [16] S. DeBacker, Parametrizing nilpotent orbits via Bruhat-Tits theory, Ann. of Math. (2), 156 (2002), 295-332. doi: 10.2307/3597191. [17] J. M. Diwadkar, Nilpotent conjugacy classes in $p$-adic Lie algebras: The odd orthogonal case, Canadian J. Math., 60 (2008), 88-108. doi: 10.4153/CJM-2008-004-6. [18] J. M. Diwadkar, Nilpotent Conjugacy Classes Of Reductive $p$-adic Lie Algebras and Definability in Pas's Language, Ph.D. Thesis, University of Pittsburgh, 2006. [19] S. Frechette, G. Gordon and L. Robson, Shalika germs for $\mathfrak{sl}_n$ and $\mathfrak{sp}_{2n}$ are motivic,, submitted., (). [20] J. Gordon, Motivic nature of character values of depth-zero representations, Int. Math. Res. Not., (2004), 1735-1760. doi: 10.1155/S1073792804133382. [21] B. Gross, On the motive of a reductive group, Invent. Math., 130 (1997), 287-313. doi: 10.1007/s002220050186. [22] T. C. Hales, Orbital integrals are motivic, Proc. Amer. Math. Soc., 133 (2005), 1515-1525. doi: 10.1090/S0002-9939-04-07740-8. [23] Harish-Chandra, Admissible Invariant Distributions on Reductive $p$-adic Groups, University Lecture Series, 16, American Mathematical Society, 1999. [24] J. C. Jantzen, Nilpotent orbits in representation theory, in Lie Theory, Progr. Math., 288, Birkhäuser Boston, Boston, MA, 2004, 1-211. [25] R. E. Kottwitz, Harmonic analysis on reductive $p$-adic groups and Lie algebras, in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Math. Proc., 4, Amer. Math. Soc., Providence, RI, 2005, 393-522. [26] E. Lawes, Motivic Integration and the Regular Shalika Germ, Ph.D. Thesis, University of Michigan, 2003. [27] G. J. McNinch, Nilpotent orbits over ground fields of good characteristic, Math. Ann., 329 (2004), 49-85. doi: 10.1007/s00208-004-0510-9. [28] A. Moy and G. Prasad, Unrefined minimal $K$-types for $p$-adic groups, Invent. Math., 116 (1994), 393-408. doi: 10.1007/BF01231566. [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), 161-190. doi: 10.1007/s10468-009-9182-1. [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), 225-233. doi: 10.1080/014453409108837187. [31] L. Robson, Shalika Germs Are Motivic, M.Sc. Essay, University of British Columbia, 2012. [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,, , (). [33] T. A. Springer, Linear Algebraic Groups, Progress in Mathematics, 9, Birkhäuser, Boston Inc, 1998. [34] J.-L. Waldspurger, Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés, Astérisque, No. 269 (2001). [35] A. Weil, Adeles and Algebraic Groups, Progress in Mathematics, 23, Birkhäuser Basel, 1982. [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), 1051-1094. doi: 10.1090/S0894-0347-96-00216-0. [37] Z. Yun, The fundamental lemma of Jacquet and Rallis, with an appendix by J. Gordon, Duke Math. J., 156 (2011), 167-227. doi: 10.1215/00127094-2010-210.
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