Shalika germs for tamely ramified elements in $GL_n$

TL;DR

This paper provides a combinatorial formula for Shalika germs of tamely ramified elements in GL_n, linking representation theory, geometry, and knot invariants, with new conjectures and polynomiality results.

Contribution

It introduces a combinatorial approach to Shalika germs for tamely ramified elements and proposes a geometric interpretation via Hilbert schemes, connecting multiple mathematical areas.

Findings

Computed weight polynomials of affine Springer fibers in type A.

Established polynomiality of point-counts of compactified Jacobians.

Provided evidence for the ORS conjecture relating Jacobians and knot invariants.

Abstract

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 $GL_n$ 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.