Shalika germs for sl(n) and sp(2n) are motivic

TL;DR

This paper demonstrates that Shalika germs for sl(n) and sp(2n) are motivic functions and establishes a uniform bound on these germs using model-theoretic methods and Harish-Chandra's theorem.

Contribution

It proves the motivic nature of Shalika germs for specific Lie algebras and provides a uniform bound, connecting harmonic analysis with model theory.

Findings

Shalika germs are motivic functions for sl(n) and sp(2n).

Established a uniform bound of q^a for normalized Shalika germs.

Connected Harish-Chandra's boundedness with motivic function bounds.

Abstract

We prove that Shalika germs on the Lie algebras sl(n) and sp(2n) belong to the class of so-called `motivic functions' defined by means of a first-order language of logic. We also prove, for these Lie algebras, a uniform bound of the form q^a (where q is the cardinality of the residue field) for the normalized Shalika germs. Our proof of the bound uses the theorem of Harish-Chandra that normalized Shalika germs are bounded, and a model-theoretic statement for uniform bounds of motivic functions from Appendix B to [arXiv:1208.1945].