Let be a complex reductive Lie algebra with Cartan algebra . Hotta and Kashiwara defined a holonomic -module , on , called the Harish-Chandra module. We relate , an associated graded module with respect to a canonical Hodge filtration on , to the isospectral commuting variety, a subvariety of which is a ramified cover of the variety of pairs of commuting elements of . Our main result establishes an isomorphism of with the structure sheaf of , the normalization of the isospectral commuting variety. We deduce, using Saito’s theory of Hodge -modules, that the scheme is Cohen–Macaulay and Gorenstein. This confirms a conjecture of M. Haiman.
Associated with any principal nilpotent pair in there is a finite subscheme of . 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 -dimensional space generated by a remarkable -harmonic polynomial on . In the special case where the above algebras are closely related to the -theorem of Haiman, and our -harmonic polynomial reduces to the Garsia–Haiman polynomial. Furthermore, in the -case, the sheaf gives rise to a vector bundle on the Hilbert scheme of points in 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.
"Isospectral commuting variety, the Harish-Chandra -module, and principal nilpotent pairs." Duke Math. J. 161 (11) 2023 - 2111, 15 August 2012. https://doi.org/10.1215/00127094-1699392