  One of the most mysterious aspects of Saito's theory of Hodge modules are the
Hodge and weight filtrations that accompany the pushforward of a Hodge module
under an open embedding. In this paper we consider the open embedding in a
product of complementary Grassmannians given by pairs of transverse subspaces.
The push-forward of the structure sheaf under this open embedding is an
important Hodge module from the viewpoint of geometric representation theory
and homological knot invariants.
  We compute the associated graded of this push-forward with respect to the
induced Hodge filtration as well as the resulting weight filtration. The main
tool is a categorical $\sl_2$ action on the category of $\D_h$-modules on
Grassmannians. Along the way we also clarify the interaction of kernels for
$\D_h$-modules with the associated graded functor. Both of these results may be
of independent interest.

Algebraic Geometry;Representation Theory