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 = 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 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)}.