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Regular version of the site

Article

On mirabolic D-modules

International Mathematics Research Notices. 2010. No. 15. P. 2947-2986.
Finkelberg M. V., Ginzburg V.

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 Graphic 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 Graphic, where Graphic denotes the flag manifold for SLn. We further relate 𝒟-modules on Graphic to 𝒟-modules on the Cartesian product SLn × ℙn−1 via a pair Graphic, 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.