Unramified affine Springer fibers and isospectral Hilbert schemes

For any connected reductive group G over C , we revisit Goresky–Kottwitz–MacPherson’s description of the torus equivariant Borel–Moore homology of affine Springer fibers Sp γ ⊂ Gr G , where γ = z t d and z is a regular semisimple element in the Lie algebra of G . In the case G = G L n , we relate the equivariant cohomology of Sp γ 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 } .