## 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 ∑ 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.

Original language | English |
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Pages (from-to) | 2947-2986 |

Number of pages | 40 |

Journal | International Mathematics Research Notices |

Volume | 2010 |

Issue number | 15 |

DOIs | |

Publication status | Published - 2010 |

Externally published | Yes |