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 γ of GLn​ 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 γ correspond to residues of torus
localization weights of a certain quasi-coherent sheaf Fγ​ on
the Hilbert scheme of points on A2, 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.Comment: 47 pages, added clarifications on the unramified case and an
application to components of affine Springer fibers, fixed typos and
reference