On Mirabolic -modules

Abstract

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 in D generated by the Lie algebra g, of 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 where the group G=SL(n) acts diagonally on X = F \times F \times P^{n-1}, a triple product where F is the flag manifold for SL(n) and P^{n-1} is the projective space. We further relate D-modules on F \times F \times P^{n-1} to D-modules on the space SL(n) \times P^{n-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.