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

Abstract

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.