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  • Victor Ginzburg (a1) (a2) and Simon Riche (a1) (a2)


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.



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1.Achar, P. N., Henderson, A. and Riche, S., Geometric Satake, Springer correspondence, and small representations II, preprint arXiv:1205.5089 (2012).
2.Achar, P. N. and Riche, S., Constructible sheaves on affine Grassmannians and geometry of the dual nilpotent cone, Israel J. Math. (2011), preprint arXiv:1102.2821 (to appear).
3.Arkhipov, S., Bezrukavnikov, R. and Ginzburg, V., Quantum groups, the loop Grassmannian, and the Springer resolution, J. Amer. Math. Soc. 17 (2004), 595678.
4.Artin, M., Tate, J. and Van den Bergh, M., Some algebras associated to automorphisms of elliptic curves, in The Grothendieck Festschrift, Vol. I, Progr. Math., Volume 86, pp. 3385 (Birkhäuser, 1990).
5.Baumann, P., Propriétés et combinatoire des bases de type canonique, mémoire d’habilitation, available on
6.Beilinson, A. and Drinfeld, V., Quantization of Hitchin’s integrable system and Hecke eigensheaves, preprint available at∼mitya/langlands.html.
7.Bernstein, I., Gel’fand, I. and Gel’fand, S., Differential operators on the base affine space and a study of g-modules, in Lie Groups and Their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), pp. 2164 (Halsted, 1975).
8.Bezrukavnikov, R., Braverman, A. and Positselskii, L., Gluing of abelian categories and differential operators on the basic affine space, J. Inst. Math. Jussieu 1 (2002), 543557.
9.Bezrukavnikov, R. and Finkelberg, M., Equivariant Satake category and Kostant–Whittaker reduction, Mosc. Math. J. 8 (2008), 3972.
10.Bezrukavnikov, R., Finkelberg, M. and Mirković, I., Equivariant homology and K-theory of affine Grassmannians and Toda lattices, Compos. Math. 141 (2005), 746768.
11.Bezrukavnikov, R. and Riche, S., Affine braid group actions on Springer resolutions, Ann. Sci. Éc. Norm. Supér. 45 (2012), 535599.
12.Braverman, A. and Finkelberg, M., Dynamical Weyl groups and equivariant cohomology of transversal slices in affine Grassmannians, Math. Res. Lett. 18 (2011), 505512.
13.Brion, M. and Kumar, S., Frobenius splitting methods in geometry and representation theory, Progr. Math., Volume 231 (Birkhäuser, 2004).
14.Broer, B., Line bundles on the cotangent bundle of the flag variety, Invent. Math. 113 (1993), 120.
15.Brylinski, R., Limits of weight spaces, Lusztig’s q-analogs, and fiberings of adjoint orbits, J. Amer. Math. Soc. 2 (1989), 517533.
16.Brylinski, J. L., Malgrange, B. and Verdier, J. L., Transformation de Fourier géométrique. II, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), 193198.
17.Deligne, P. and Milne, J., Tannakian categories, in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Math., Volume 900 (Springer, 1982).
18.Dodd, C., Equivariant coherent sheaves, Soergel bimodules, and categorification of affine Hecke algebras, preprint arXiv:1108.4028 (2011).
19.Etingof, P. and Varchenko, A., Dynamical Weyl groups and applications, Adv. Math. 167 (2002), 74127.
20.Evens, S. and Mirković, I., Fourier transform and the Iwahori–Matsumoto involution, Duke Math. J. 86 (1997), 435464.
21.Fiebig, P. and Williamson, G., Parity sheaves, moment graphs and the $p$-smooth locus of Schubert varieties, Ann. Inst. Fourier (Grenoble), preprint arXiv:1008.0719, 2010 (to appear).
22.Ginzburg, V., Perverse sheaves on a loop group and Langlands’ duality, preprint arXiv:alg-geom/9511007 (1995).
23.Ginzburg, V., Variations on themes of Kostant, Transform. Groups 13 (2008), 557573.
24.Ginzburg, V. and Kazhdan, D., A class of symplectic varieties associated with the space $G/U$ (in preparation).
25.Humphreys, J. E., Representations of Semisimple Lie Algebras in the BGG Category O, Graduate Studies in Mathematics, Volume 94 (American Mathematical Society, Providence, RI, 2008).
26.Jantzen, J. C., Representations of algebraic groups, second edition, Mathematical Surveys and Monographs, Volume 107 (Amer. Math. Soc., 2003).
27.Jantzen, J. C., Nilpotent orbits in representation theory, in Lie theory, Progr. Math., Volume 228, pp. 1211 (Birkhäuser, 2004).
28.Kashiwara, M., The universal Verma module and the b-function, in Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math., Volume 6, pp. 6781 (North-Holland, 1985).
29.Kazhdan, D. and Laumon, G., Gluing of perverse sheaves and discrete series representation, J. Geom. Phys. 5 (1988), 63120.
30.Kazhdan, D. and Lusztig, G., Schubert varieties and Poincaré duality, in Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., Volume XXXVI, pp. 185203 (Amer. Math. Soc., 1980).
31.Kostant, B., The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 9731032.
32.Levasseur, T. and Stafford, J. T., Differential operators and cohomology groups on the basic affine space, in Studies in Lie Theory, Progr. Math., Volume 243, pp. 377403 (Birkhäuser, 2006).
33.Lusztig, G., Cuspidal local systems and graded Hecke algebras. I, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 145202.
34.Matsumura, H., Commutative Ring Theory (Cambridge University Press, 1986).
35.Mirković, I. and Vilonen, K., Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. 166 (2007), 95143.
36.Ngô, B. C. and Polo, P., Résolutions de Demazure affines et formule de Casselman–Shalika géométrique, J. Algebraic Geom. 10 (2001), 515547.
37.Shapovalov, N., Structure of the algebra of differential operators on the basic affine space, Funct. Anal. Appl. 8 (1974), 3746.
38.Springer, T., Quelques applications de la cohomologie d’intersection, Astérisque 92–93 (1982), 249273.
39.Tarasov, V. and Varchenko, A., Difference equations compatible with trigonometric KZ differential equations, Int. Math. Res. Not. 2000(15) 801829.
40.Vasserot, E., On the action of the dual group on the cohomology of perverse sheaves on the affine Grassmannian, Compos. Math. 131 (2002), 5160.
41.Yekutieli, A. and Zhang, J. J., Dualizing complexes and tilting complexes over simple rings, J. Algebra 256 (2002), 556567.
42.Yun, Z. and Zhu, X., Integral homology of loop groups via Langlands dual group, Represent. Theory 15 (2011), 347369.
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  • Victor Ginzburg (a1) (a2) and Simon Riche (a1) (a2)


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