Unramified affine Springer fibers and isospectral Hilbert schemes

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 $\gamma =a{t}^d$, and $a$ is a regular semisimple element in the Lie algebra of $G$. In the case $G=G{L}_n$, 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 $\{x^n=y^{dn}\}$.

42 pages, section 3 rewritten due to an erroneous lemma in the previous version