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 generated by the Lie algebra 𝔤 = LieG. 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 = SLn acting diagonally on , where denotes the flag manifold for SLn. We further relate 𝒟-modules on to 𝒟-modules on the Cartesian product SLn × ℙn−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.