Shalika Germs for \(\mathfrak{s}\mathfrak{l}_{n}\) and \(\mathfrak{s}\mathfrak{p}_{2n}\) Are Motivic

Conference paper
Part of the Association for Women in Mathematics Series book series (AWMS, volume 2)


We prove that Shalika germs on the Lie algebras \(\mathfrak{s}\mathfrak{l}_{n}\) and \(\mathfrak{s}\mathfrak{p}_{2n}\) belong to the class of so-called motivic functions defined by means of a first-order language of logic. It is a well-known theorem of Harish-Chandra that for a Lie algebra \(\mathfrak{g}(F)\) over a local field F of characteristic zero, the Shalika germs, normalized by the square root of the absolute value of the discriminant, are bounded on the set of regular semisimple elements \(\mathfrak{g}^{\mathrm{rss}}\), however, it is not easy to see how this bound depends on the field F. As a consequence of the fact that Shalika germs are motivic functions for \(\mathfrak{s}\mathfrak{l}_{n}\) and \(\mathfrak{s}\mathfrak{p}_{2n}\), we prove that for these Lie algebras, this bound must be of the form qa, where q is the cardinality of the residue field of F, and a is a constant. Our proof that Shalika germs are motivic in these cases relies on the interplay of DeBacker’s parametrization of nilpotent orbits with the parametrization using partitions, and the explicit matching between these parametrizations due to Nevins (Algebra Representation Theory 14, 161–190, 2011). We include two detailed examples of the matching of these parametrizations.


  1. Barbasch, D., Moy, A.: Local character expansions. Ann. Sci. École Norm. Sup. (4) 30(5), 553–567 (1997) (English, with English and French summaries). MR1474804 (99j:22021)Google Scholar
  2. Cluckers, R., Loeser, F.: Constructible motivic functions and motivic integration. Invent. Math. 173(1), 23–121 (2008)MATHMathSciNetCrossRefGoogle Scholar
  3. Cluckers, R., Loeser, F.: Motivic integration in all residue field characteristics for Henselian discretely valued fields of characteristic zero. J. Reine Angew. Math. 701, pp. 1–31, (2015)MATHMathSciNetGoogle Scholar
  4. Cluckers, R., Gordon, J., Halupczok, I.: Local integrability results in harmonic analysis on reductive groups in large positive characterisitic. Ann. Sci. Ec. Norm. Sup. 47(6) pp. 1163–1195 (2014a)
  5. Cluckers, R., Gordon, J., Halupczok, I.: Motivic functions, integrability, and applications to harmonic analysis on p-adic groups. Electron. Res. Announc. (2014b). 21, pp. 137–152, (2014)
  6. Cluckers, R., Hales, T., Loeser, F.: Transfer principle for the fundamental lemma. In: Clozel, L., Harris, M., Labesse, J.-P., Ngô, B.-C. (eds.) On the Stabilization of the Trace Formula. International Press, Boston (2011b)Google Scholar
  7. Collingwood, D.H., McGovern, W.M.: Nilpotent Orbits in Semisimple Lie Algebras. Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York (1993). MR1251060 (94j:17001)Google Scholar
  8. DeBacker, S.: Homogeneity results for invariant distributions of a reductive p-adic group. Ann. Sci. École Norm. Sup. (4) 35(3), 391–422 (2002a) (English, with English and French summaries). MR1914003 (2003i:22019)Google Scholar
  9. DeBacker, S.: Parametrizing nilpotent orbits via Bruhat-Tits theory. Ann. Math. (2) 156(1), 295–332 (2002b). MR1935848 (2003i:20086)Google Scholar
  10. Diwadkar, J.M.: Nilpotent conjugacy classes of reductive p-adic Lie algebras and definability in Pas’s language. Ph.D. Thesis, University of Pittsburgh (2006)Google Scholar
  11. Harish-Chandra: Harmonic analysis on reductive p-adic groups. In: Moore, C.C. (ed.) Harmonic Analysis on Homogeneous Spaces. Proceedings of Symposia in Pure Mathematics, vol. 26, pp. 167–192. American Mathematical Society, Providence (1973). MR0340486 (49 #5238)Google Scholar
  12. Harish-Chandra: Admissible invariant Distributions on Reductive p-Adic Groups. University Lecture Series, vol. 16. American Mathematical Society, Providence (1999). With a preface and notes by Stephen DeBacker and Paul J. Sally, Jr. MR1702257 (2001b:22015)Google Scholar
  13. Kottwitz, R.E.: Harmonic analysis on reductive p-adic groups and Lie algebras. In: Harmonic Analysis, the Trace Formula, and Shimura Varieties, pp. 393–522 (2005). MR2192014, American mathematical Society, Providence, RI for Clay Mathematics Institute, Cambridge, MA, (2006m:22016)Google Scholar
  14. Lam, T.Y.: Introduction to Quadratic Forms Over Fields. Graduate Studies in Mathematics, vol. 67. American Mathematical Society, Providence (2005)Google Scholar
  15. McNinch, G.J.: Nilpotent orbits over ground fields of good characteristic. Math. Ann. 329(1), 49–85 (2004). MR2052869 (2005j:17018)Google Scholar
  16. Moy, A., Prasad, G.: Unrefined minimal K-types for p-adic groups. Invent. Math. 116(1–3), 393–408 (1994). MR1253198 (95f:22023)Google Scholar
  17. Nevins, M.: On nilpotent orbits of \(\mathrm{SL}_{n}\) and Sp2n over a local non-Archimedean field. Algebra Representation Theory 14, 161–190 (2011)MATHMathSciNetCrossRefGoogle Scholar
  18. Rabinoff, J.: The Bruhat-Tits building of a p-adic Chevalley group and an application to representation theory. Harvard Senior Thesis (2005)Google Scholar
  19. Ranga Rao, R.: Orbital integrals in reductive groups. Ann. Math. (2) 96, 505–510 (1972). MR0320232 (47 #8771)Google Scholar
  20. Robson, L.: Shalika Germs are Motivic. M.Sc. Essay, University of British Columbia, Vancouver (2012)Google Scholar
  21. Shalika, J.A.: A theorem on semi-simple \(\mathcal{P}\)-adic groups. Ann. Math. (2) 95, 226–242 (1972). MR0323957 (48 #2310)Google Scholar
  22. Shin, S.W., Templier, N.: Sato-Tate Theorem for Families and Low-Lying Zeroes of Automorphic L-functions. Preprint. With appendices by R. Kottwitz and J. Gordon, R. Cluckers and I. Halupczok, Invent. Math. (2015)
  23. Waldspurger, J.-L.: Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés. Astérisque 269, vi+449 (2001)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Sharon M. Frechette
    • 1
  • Julia Gordon
    • 2
  • Lance Robson
    • 2
  1. 1.Dept. of Mathematics & Computer ScienceCollege of the Holy CrossWorcesterUSA
  2. 2.Mathematics DepartmentThe University of British ColumbiaVancouverCanada

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