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

TL;DR

This paper establishes a deep connection between the Harish-Chandra D-module, the isospectral commuting variety, and principal nilpotent pairs, revealing their geometric and algebraic properties and confirming conjectures about their structure.

Contribution

It proves the isomorphism between the associated graded of the Harish-Chandra module and the normalization of the isospectral commuting variety, confirming conjectures on their Cohen-Macaulay and Gorenstein properties.

Findings

Normalization of the isospectral commuting variety is Cohen-Macaulay and Gorenstein.

Associated coordinate rings form bigraded Gorenstein algebras with Weyl group symmetry.

In the gl_n case, results relate to the n!-theorem and Garsia-Haiman polynomials.

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.