Shalika germs for tamely ramified elements in $G{L}_n$

Degenerating the action of the elliptic Hall algebra on the Fock space, we give a combinatorial formula for the Shalika germs of tamely ramified regular semisimple elements $\gamma$ of $G{L}_n$ over a nonarchimedean local field. As a byproduct, we compute the weight polynomials of affine Springer fibers in type A and orbital integrals of tamely ramified regular semisimple elements. We conjecture that the Shalika germs of $\gamma$ correspond to residues of torus localization weights of a certain quasi-coherent sheaf ${\mathcal{F}}_{\gamma }$ on the Hilbert scheme of points on ${\mathbb{A}}^2$, thereby finding a geometric interpretation for them. As corollaries, we obtain the polynomiality in $q$ of point-counts of compactified Jacobians of planar curves, as well as a virtual version of the Cherednik-Danilenko conjecture on their Betti numbers. Our results also provide further evidence for the ORS conjecture relating compactified Jacobians and HOMFLY-PT invariants of algebraic knots.