Isospectral commuting variety and the Harish-Chandra D-module

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 give an explicit description of gr(M), the associated graded module with respect to a canonical Hodge filtration on M. The description involves the isospectral commuting variety, a subvariety X of g x g x h x h which is a finite extension of the variety of pairs of commuting elements of g. Our main result establishes an isomorphism of gr(M) with the structure sheaf of the normalization of X. It follows, thanks to the theory of polarized Hodge modules, that the normalization of the isospectral commuting variety is Cohen-Macaulay and Gorenstein, confirming a conjecture of M. Haiman. In the special case where g=gl_n, there is an open subset of the isospectral commuting variety that is closely related to the Hilbert scheme of n points in the plane. The sheaf gr(M) gives rise to a locally free sheaf on the Hilbert scheme. We show that the corresponding vector bundle is isomorphic to the Procesi bundle that plays an important role in the work of M. Haiman.