Affine Springer Fibers and Generalized Haiman Ideals

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.

33 pages, with an appendix by Eugene Gorsky and Joshua P. Turner

References in corpus (3)

• Affine pavings for affine Springer fibers for split elements in PGL(3)
• The affine Springer fiber-sheaf correspondence
• Splitting maps in link Floer homology and integer points in permutahedra

Cited by in corpus (1)

• The Hilb-vs-Quot Conjecture