  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.
