On mirabolic D-modules

Michael Finkelberg, Victor Ginzburg

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)

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 = SLn acting diagonally on X = B × B × ℙ n-1, where B denotes the flag manifold for SLn. We further relate D-modules on to B × B × ℙn-1 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.

Original languageEnglish
Pages (from-to)2947-2986
Number of pages40
JournalInternational Mathematics Research Notices
Volume2010
Issue number15
DOIs
Publication statusPublished - 2010
Externally publishedYes

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