## Duke Mathematical Journal

### Isospectral commuting variety, the Harish-Chandra $\mathbf{\mathcal{D}}$-module, and principal nilpotent pairs

Victor Ginzburg

#### Abstract

Let $\mathfrak {g}$ be a complex reductive Lie algebra with Cartan algebra ${\mathfrak{t}}$. Hotta and Kashiwara defined a holonomic $\mathscr{D}$-module ${\mathcal{M}}$, on $\mathfrak {g}\times{\mathfrak{t}}$, called the Harish-Chandra module. We relate $\operatorname {gr}{\mathcal{M}}$, 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 $\operatorname {gr}{\mathcal{M}}$ with the structure sheaf of ${\mathfrak{X}}_{{\operatorname {norm}}}$, the normalization of the isospectral commuting variety. We deduce, using Saito’s theory of Hodge ${\mathscr{D}}$-modules, that the scheme ${\mathfrak{X}}_{{\operatorname {norm}}}$ 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 ${\mathfrak{X}}_{{\operatorname {norm}}}$. 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 $\operatorname {gr}{\mathcal{M}}$ 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.

#### Article information

Source
Duke Math. J., Volume 161, Number 11 (2012), 2023-2111.

Dates
First available in Project Euclid: 24 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1343133924

Digital Object Identifier
doi:10.1215/00127094-1699392

Mathematical Reviews number (MathSciNet)
MR2957698

Zentralblatt MATH identifier
1271.14064

#### Citation

Ginzburg, Victor. Isospectral commuting variety, the Harish-Chandra $\mathbf{\mathcal{D}}$ -module, and principal nilpotent pairs. Duke Math. J. 161 (2012), no. 11, 2023--2111. doi:10.1215/00127094-1699392. https://projecteuclid.org/euclid.dmj/1343133924

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