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 D g ∑ D generated by the Lie algebra g = Lie G. Then, D/D g 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/D g 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 = SL_n acting diagonally on X = B × B × ℙ ^n-1, where B denotes the flag manifold for SL_n. We further relate D-modules on to B × B × ℙ^n-1 to D-modules on the Cartesian product SL_n × ℙ^n-1 via a pair, 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 PauseMathClassHC gives an exact functor on the abelian category of mirabolic D-modules.