On Mirabolic D-modules

Let an algebraic group G act on X, a connected algebraic manifold, with finitely many orbits. For any Harish-Chandra pair (D, G) where D is a sheaf of twisted differential operators on X, we form a left ideal Dg subset of D generated by the Lie algebra g = Lie G. Then, D/Dg is a holonomic D-module, and its restriction to a unique Zariski open dense G-orbit in X is a G-equivariant local system. We prove a criterion saying that the D-module D/Dg is isomorphic, under certain (quite restrictive) conditions, to a direct image of that local system to X. We apply this criterion in the special case of the group G = SLn acting diagonally on X = B x B x Pn-1, where B denotes the flag manifold for SLn. We further relate D-modules on B x B x Pn-1 to D-modules on the Cartesian product SLn x Pn-1 via a pair (CH, HC), of adjoint functors analogous to those used in Lusztig's theory of character sheaves. A second important result of the paper provides an explicit description of these functors, showing that the functor HC gives an exact functor on the abelian category of mirabolic D-modules.