Unramified affine Springer fibers and isospectral Hilbert schemes

TL;DR

This paper explores the structure of affine Springer fibers for reductive groups, relating their equivariant cohomology to isospectral Hilbert schemes and knot homology, revealing deep geometric and algebraic connections.

Contribution

It establishes a link between affine Springer fibers' equivariant cohomology and isospectral Hilbert schemes, extending to connections with HOMFLY homology and formulating a new conjecture.

Findings

Relation between affine Springer fibers and isospectral Hilbert schemes for GL_n

Connection to HOMFLY homology of torus links

Proposed conjecture on homology of Hilbert schemes on singular curves

Abstract

For any connected reductive group $G$ over $\mathbb{C}$, we revisit Goresky-Kottwitz-MacPherson's description of the torus equivariant Borel-Moore homology of affine Springer fibers $\mathrm{Sp}_\gamma\subset \mathrm{Gr}_G$, where $\gamma=at^d$, and $a$ is a regular semisimple element in the Lie algebra of $G$. In the case $G = GL_n$, we relate the equivariant cohomology of $\mathrm{Sp}_\gamma$ to Haiman's work on the isospectral Hilbert scheme of points on the plane. We also explain the connection to the HOMFLY homology of $(n, dn)$-torus links, and formulate a conjecture describing the homology of the Hilbert scheme of points on the curve $\{x^n=y^{dn}\}$.