Old and new geometry for Shalika germs

Abstract: In recent work with Tsai, we were able to give a completely combinatorial formula for so called Shalika germs of tamely ramified regular semisimple elements in GL_n over a non-archimedean local field F. This was done using methods from harmonic analysis, representation theory, and rather surprisingly, knot theory. 

Shalika germs are a family of motivic functions on Lie(GL_n(F)) indexed by nilpotent orbits(=partitions), which refine e.g. regular semisimple orbital integrals, point-counts of compactified Jacobians of plane curve singularities, and supercuspidal character values of the p-adic group (none of which have been well understood in the last 40-50 years). 

I will give an introduction to Shalika germs and explain their relation to point-counting on compactified Jacobians. I will then give some examples and outline our method of computation, which reveals an intimate connection to the Hilbert scheme of points on C^2.