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 \(\gamma =zt^d\) and z 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 \(\{x^n=y^{dn}\}\).