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

Abstract

Let $\mathfrak{g}$ be a complex reductive Lie algebra with Cartan algebra $\mathfrak{t}$. Hotta and Kashiwara defined a holonomic $\mathcal{D}$-module $\mathcal{M}$, on $\mathfrak{g}\times \mathfrak{t}$, called the Harish-Chandra module. We relate $gr\mathcal{M}$, an associated graded module with respect to a canonical Hodge filtration on $\mathcal{M}$, to the isospectral commuting variety, a subvariety of $\mathfrak{g}\times \mathfrak{g}\times \mathfrak{t}\times \mathfrak{t}$ which is a ramified cover of the variety of pairs of commuting elements of $\mathfrak{g}$. Our main result establishes an isomorphism of $gr\mathcal{M}$ with the structure sheaf of ${\mathfrak{X}}_{norm}$, the normalization of the isospectral commuting variety. We deduce, using Saito’s theory of Hodge $\mathcal{D}$-modules, that the scheme ${\mathfrak{X}}_{norm}$ is Cohen–Macaulay and Gorenstein. This confirms a conjecture of M. Haiman.

Associated with any principal nilpotent pair in $\mathfrak{g}$ there is a finite subscheme of ${\mathfrak{X}}_{norm}$. 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 space generated by a remarkable $W$-harmonic polynomial on $\mathfrak{t}\times \mathfrak{t}$. In the special case where $\mathfrak{g}=\mathfrak{g}{\mathfrak{l}}_n$ 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 $\mathfrak{g}{\mathfrak{l}}_n$-case, the sheaf $gr\mathcal{M}$ gives rise to a vector bundle on the Hilbert scheme of $n$ points in ${\mathbb{C}}^2$ 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.