Weight polynomials of compactified Jacobians and link invariants

Abstract: Using a recursive algorithm for orbital integrals of tamely ramified elliptic elements in p-adic GL_n, we compute the weight polynomials of (local) compactified Jacobians of planar curves. Depending on one's taste, these can be also interpreted as point-counts on Hitchin fibers or affine Springer fibers in type A. The algorithm is based on old work of Waldspurger and can be interpreted using an action of the affine Yangian of gl(1) on the Fock space, where it becomes clear that there is a relationship to knot invariants of HOMFLY type. This also proves a virtual version of the Cherednik-Danilenko conjecture on Betti numbers of Jacobian factors. In fact, the algorithm yields more, such as so called Shalika germs for the elements in question. These have a geometric interpretation on the Hilbert scheme of points on C^2, which I will also discuss. This is joint work with Cheng-Chiang Tsai.

algebraic geometrysymplectic geometry

Audience: researchers in the topic

M-seminar