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DIFFERENTIAL OPERATORS ON $G/U$ AND THE AFFINE GRASSMANNIAN

Published online by Cambridge University Press:  07 April 2014

Victor Ginzburg  and
Simon Riche
Victor Ginzburg
Affiliation:
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA (ginzburg@math.uchicago.edu) Université Blaise Pascal - Clermont-Ferrand II, Laboratoire de Mathématiques, CNRS, UMR 6620, Campus universitaire des Cézeaux, F-63177 Aubière Cedex, France (simon.riche@math.univ-bpclermont.fr)
Simon Riche
Affiliation:
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA (ginzburg@math.uchicago.edu) Université Blaise Pascal - Clermont-Ferrand II, Laboratoire de Mathématiques, CNRS, UMR 6620, Campus universitaire des Cézeaux, F-63177 Aubière Cedex, France (simon.riche@math.univ-bpclermont.fr)
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Abstract

We describe the equivariant cohomology of cofibers of spherical perverse sheaves on the affine Grassmannian of a reductive algebraic group in terms of the geometry of the Langlands dual group. In fact we give two equivalent descriptions: one in terms of $\mathscr{D}$-modules of the basic affine space, and one in terms of intertwining operators for universal Verma modules. We also construct natural collections of isomorphisms parameterized by the Weyl group in these three contexts, and prove that they are compatible with our isomorphisms. As applications we reprove some results of the first author and of Braverman and Finkelberg.

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 14 , Issue 3 , July 2015 , pp. 493 - 575
Copyright
© Cambridge University Press 2014 

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