Speaker: Anton Mellit - University of Vienna

Title: Affine Springer fibers, open Hessenberg varieties, and nabla positivity.

Abstract: I will talk about the positive part of a certain affine Springer fiber studied by Goresky, Kottwitz, and MacPherson, and a certain interesting open subvariety. The Hilbert series of their Borel-Moore homology turns out to be related to reproducing kernels of the Bergeron-Garsia nabla operator. This operator is easy to define in the basis of modified Macdonald polynomials, but producing explicit combinatorial evaluations of this operator is usually difficult and (conjecturally) relates to interesting Hilbert series associated to various moduli spaces. Our work is motivated by the nabla positivity conjecture of Bergeron, Garsia, Haiman, and Tesler that predicts that nabla evaluated on a Schur function is sometimes positive, sometimes negative. We categorify this conjecture and reduce it to a vanishing conjecture for the interesting open variety. It turns out, each irreducible S_n representation mysteriously prefers to live in certain degrees and weights in the cohomology. This is a joint work with Erik Carlsson.