Isospectral commuting variety, the Harish-Chandra D-module, and principal nilpotent pairs

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Let g be a complex reductive Lie algebra with Cartan algebra t. Hotta and Kashiwara defined a holonomic D-module M, on g×t, called the Harish-Chandra module. We relate grM, an associated graded module with respect to a canonical Hodge filtration on M, to the isospectral commuting variety, a subvariety of g×g×t×t which is a ramified cover of the variety of pairs of commuting elements of g. Our main result establishes an isomorphism of grM with the structure sheaf of Xnorm, the normalization of the isospectral commuting variety. We deduce, using Saito’s theory of Hodge D-modules, that the scheme Xnorm is Cohen–Macaulay and Gorenstein. This confirms a conjecture of M. Haiman. Associated with any principal nilpotent pair in g there is a finite subscheme of Xnorm. The corresponding coordinate ring is a bigraded finite-dimensional Gorenstein algebra that affords the regular representation of the Weyl group. The socle of that algebra is a 1-dimensional space generated by a remarkable W-harmonic polynomial on t×t. In the special case where g=gln the above algebras are closely related to the n!-theorem of Haiman, and our W-harmonic polynomial reduces to the Garsia–Haiman polynomial. Furthermore, in the gln-case, the sheaf grM gives rise to a vector bundle on the Hilbert scheme of n points in C2 that turns out to be isomorphic to the Procesi bundle. Our results were used by I. Gordon to obtain a new proof of positivity of the Kostka–Macdonald polynomials established earlier by Haiman.