Affine Springer Fibers and Generalized Haiman Ideals

TL;DR

This paper computes the homology of affine Springer fibers for GL_n, relates them to Haiman ideals, and connects these to link homology and generalized Catalan numbers, providing explicit descriptions and proving a conjecture.

Contribution

It establishes a link between affine Springer fibers, Haiman ideals, and link homology, with explicit computations for n=3 and verification of a conjecture.

Findings

Computed Borel-Moore homology of affine Springer fibers for GL_n.

Explicitly described Haiman ideals for n=3, including Hilbert series and generators.

Connected homology of affine Springer fibers to Khovanov-Rozansky homology and generalized Catalan numbers.

Abstract

We compute the Borel-Moore homology of unramified affine Springer fibers for $\mathrm{GL}_n$ under the assumption that they are equivariantly formal and relate them to certain ideals discussed by Haiman. For $n=3$, we give an explicit description of these ideals, compute their Hilbert series, generators and relations, and compare them to generalized $(q,t)$ Catalan numbers. We also compare the homology to the Khovanov-Rozansky homology of the associated link, and prove a version of a conjecture of Oblomkov, Rasmussen, and Shende in this case.