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