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We show that the Frobenius character of the equivariant Borel-Moore homologyof a certain positive $GL_n$-version of the unramified affine Springer fiber$Z_k$ studied by Goreski, Kottwitz and MacPherson is computed by the matrixcoefficients of the $\nabla^k$-operator, which acts diagonally in the modifiedMacdonald basis. We do this by relating the combinatorial formula for the$\nabla^k$-operator we obtained in an earlier paper to the GKM paving of $Z_k$,and we give an algebraic presentation of the above homology as an explicitsubmodule of the Kostant-Kumar nil Hecke algebra. We then study a certain openlocus $U_k \subset Z_k$, and reduce a long-standing conjecture of Bergeron,Garsia, Haiman and Tesler, which predicts the sign of the coefficients of theSchur expansion of $\nabla^k$, to a vanishing conjecture about the homologygroups of $U_k$. The latter conjecture is in turn reduced to a vanishingconjecture for certain open loci of the regular semisimple Hessenberg varietieswhich are indexed by partial Dyck paths.

GKM spaces, and the signed positivity of the nabla operator