Isospectral commuting variety, the Harish-Chandra D-module, and principal nilpotent pairs (1108.5367v2)

Abstract: Let g be a complex reductive Lie algebra with Cartan algebra h. Hotta and Kashiwara defined a holonomic D-module M, on g x h, called Harish-Chandra module. We relate gr(M), an associated graded module with respect to a canonical Hodge filtration on M, to the isospectral commuting variety, a subvariety of g x g x h x h which is a ramified cover of the variety of pairs of commuting elements of g. Our main result establishes an isomorphism of gr(M) with the structure sheaf of X_norm, the normalization of the isospectral commuting variety. It follows, thanks to the theory of Hodge modules, that the normalization of the isospectral commuting variety is Cohen-Macaulay and Gorenstein, confirming a conjecture of M. Haiman. We deduce, using Saito's theory of Hodge D-modules, that the scheme X_norm 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 X_norm. 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 vector space generated by a remarkable W-harmonic polynomial on h x h. In the special case where g=gl_n 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 gl_n case, the sheaf gr(M) gives rise to a vector bundle on the Hilbert scheme of n points in C^2 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.