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

Previous Article
Canonical Cartan connections on maximally minimal generic submanifolds $\mathbf{M^5 \subset \mathbb{C}^4}$
ERA-MS Home
This Volume
Next Article
An arithmetic ball quotient surface whose Albanese variety is not of CM type

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

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

Raf Cluckers ^1, , Julia Gordon ^2, and Immanuel Halupczok ^3,

1.	Université Lille 1, Laboratoire Painlevé, CNRS - UMR 8524, Cité 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

Abstract
Related Papers

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].

Keywords: Harish-Chandra character, motivic integration., p-adic group, positive characteristic, orbital integral.

Mathematics Subject Classification: 22D12 (03C98.

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.

[1]	Aihua Fan, Shilei Fan, Lingmin Liao, Yuefei Wang. Minimality of p-adic rational maps with good reduction. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3161-3182. doi: 10.3934/dcds.2017135
[2]	Farrukh Mukhamedov, Otabek Khakimov. Chaotic behavior of the P-adic Potts-Bethe mapping. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 231-245. doi: 10.3934/dcds.2018011
[3]	James Kingsbery, Alex Levin, Anatoly Preygel, Cesar E. Silva. Dynamics of the $p$-adic shift and applications. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 209-218. doi: 10.3934/dcds.2011.30.209
[4]	Alexander Kemarsky, Frédéric Paulin, Uri Shapira. Escape of mass in homogeneous dynamics in positive characteristic. Journal of Modern Dynamics, 2017, 11: 369-407. doi: 10.3934/jmd.2017015
[5]	Amir Mohammadi. Measures invariant under horospherical subgroups in positive characteristic. Journal of Modern Dynamics, 2011, 5 (2) : 237-254. doi: 10.3934/jmd.2011.5.237
[6]	Eric Férard. On the irreducibility of the hyperplane sections of Fermat varieties in $\mathbb{P}^3$ in characteristic $2$. Advances in Mathematics of Communications, 2014, 8 (4) : 497-509. doi: 10.3934/amc.2014.8.497
[7]	Sanghoon Kwon, Seonhee Lim. Equidistribution with an error rate and Diophantine approximation over a local field of positive characteristic. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 169-186. doi: 10.3934/dcds.2018008
[8]	Ilwoo Cho. Certain $$-homomorphisms acting on unital $C^{}$-probability spaces and semicircular elements induced by $p$-adic number fields over primes $p$. Electronic Research Archive, 2020, 28 (2) : 739-776. doi: 10.3934/era.2020038
[9]	T. Diogo, N. B. Franco, P. Lima. High order product integration methods for a Volterra integral equation with logarithmic singular kernel. Communications on Pure and Applied Analysis, 2004, 3 (2) : 217-235. doi: 10.3934/cpaa.2004.3.217
[10]	Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265
[11]	Mingchun Wang, Jiankai Xu, Huoxiong Wu. On Positive solutions of integral equations with the weighted Bessel potentials. Communications on Pure and Applied Analysis, 2019, 18 (2) : 625-641. doi: 10.3934/cpaa.2019031
[12]	Yutian Lei. Positive solutions of integral systems involving Bessel potentials. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2721-2737. doi: 10.3934/cpaa.2013.12.2721
[13]	Nikolaos S. Papageorgiou, George Smyrlis. Positive solutions for parametric $p$-Laplacian equations. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1545-1570. doi: 10.3934/cpaa.2016002
[14]	Friedemann Brock, Leonelo Iturriaga, Justino Sánchez, Pedro Ubilla. Existence of positive solutions for $p$--Laplacian problems with weights. Communications on Pure and Applied Analysis, 2006, 5 (4) : 941-952. doi: 10.3934/cpaa.2006.5.941
[15]	Fausto Ferrari, Qing Liu, Juan Manfredi. On the characterization of $p$-harmonic functions on the Heisenberg group by mean value properties. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2779-2793. doi: 10.3934/dcds.2014.34.2779
[16]	L. Brandolini, M. Rigoli and A. G. Setti. On the existence of positive solutions of Yamabe-type equations on the Heisenberg group. Electronic Research Announcements, 1996, 2: 101-107.
[17]	Antonio Greco, Giovanni Porru. Optimization problems for the energy integral of p-Laplace equations. Conference Publications, 2013, 2013 (special) : 301-310. doi: 10.3934/proc.2013.2013.301
[18]	Gennaro Infante. Eigenvalues and positive solutions of odes involving integral boundary conditions. Conference Publications, 2005, 2005 (Special) : 436-442. doi: 10.3934/proc.2005.2005.436
[19]	K. Q. Lan. Positive solutions of semi-Positone Hammerstein integral equations and applications. Communications on Pure and Applied Analysis, 2007, 6 (2) : 441-451. doi: 10.3934/cpaa.2007.6.441
[20]	G. Infante. Positive solutions of some nonlinear BVPs involving singularities and integral BCs. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 99-106. doi: 10.3934/dcdss.2008.1.99

2020 Impact Factor: 0.929

Tools

Metrics

Other articles
by authors

on AIMS
Raf Cluckers Julia Gordon Immanuel Halupczok
on Google Scholar
Raf Cluckers Julia Gordon Immanuel Halupczok

[Back to Top]