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

  • Sharon M. Frechette
  • Julia Gordon
  • Lance Robson
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 q a , 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.



This paper clearly owes a debt to the ideas of T.C. Hales and to the thesis of Jyotsna Diwadkar. The second author is grateful to Raf Cluckers and Immanuel Halupczok for multiple helpful communications. We thank the organizers of the WIN workshop in Luminy who made this collaboration possible. The second and third authors were supported by NSERC.


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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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