Shalika germs for tamely ramified elements in $GL_n$

Abstract

We prove explicit combinatorial formulas for various germ expansions of orbital integrals of tamely ramified elements in $GL_n(F)$, where $F$ is a nonarchimedean local field. The relevant combinatorics arises from the theory of the elliptic Hall algebra and the representation-theoretic knot superpolynomials defined by Cherednik--Danilenko and Morton--Samuelson. As a byproduct, we give explicit formulas for the weight polynomials of affine Springer fibers in type A and standard orbital integrals of tamely ramified regular semisimple elements. Our formulas subsume most earlier facts about the structure of Shalika germs of $GL_n$ in the literature. As a further corollary, we show that point-counts of compactified Jacobians of locally planar curves are given by non-negative integral polynomials. Our results also provide further evidence for the Oblomkov-Rasmussen-Shende conjecture relating compactified Jacobians and HOMFLY-PT invariants of algebraic knots.