Duke Mathematical Journal

Pursuing the double affine Grassmannian, I: Transversal slices via instantons on Ak-singularities

Alexander Braverman and Michael Finkelberg

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

This article is the first in a series that describes a conjectural analog of the geometric Satake isomorphism for an affine Kac-Moody group. (For simplicity, we only consider the untwisted and simply connected case here.) The usual geometric Satake isomorphism for a reductive group G identifies the tensor category Rep(G) of finite-dimensional representations of the Langlands dual group G with the tensor category PervG(O)(GrG) of G(O)-equivariant perverse sheaves on the affine Grassmannian GrG=G(K)/G(O) of G. (Here K=C((t)) and O=C[[t]].) As a by-product one gets a description of the irreducible G(O)-equivariant intersection cohomology (IC) sheaves of the closures of G(O)-orbits in GrG in terms of q-analogs of the weight multiplicity for finite-dimensional representations of G.

The purpose of this article is to try to generalize the above results to the case when G is replaced by the corresponding affine Kac-Moody group Gaff. (We refer to the (not yet constructed) affine Grassmannian of Gaff as the double affine Grassmannian.) More precisely, in this article we construct certain varieties that should be thought of as transversal slices to various Gaff(O)-orbits inside the closure of another Gaff(O)-orbit in GrGaff. We present a conjecture that computes the intersection cohomology sheaf of these varieties in terms of the corresponding q-analog of the weight multiplicity for the Langlands dual affine group Gaff, and we check this conjecture in a number of cases.

Some further constructions (such as convolution of the corresponding perverse sheaves, analog of the Beilinson-Drinfeld Grassmannian, and so forth) will be addressed in another publication

Article information

Source
Duke Math. J., Volume 152, Number 2 (2010), 175-206.

Dates
First available in Project Euclid: 31 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1270041107

Digital Object Identifier
doi:10.1215/00127094-2010-011

Mathematical Reviews number (MathSciNet)
MR2656088

Zentralblatt MATH identifier
1200.14083

Subjects
Primary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx]
Secondary: 14D21: Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) [See also 32L25, 81Txx]

Citation

Braverman, Alexander; Finkelberg, Michael. Pursuing the double affine Grassmannian, I: Transversal slices via instantons on $A_k$ -singularities. Duke Math. J. 152 (2010), no. 2, 175--206. doi:10.1215/00127094-2010-011. https://projecteuclid.org/euclid.dmj/1270041107


Export citation

References

  • A. Beilinson and V. Drinfeld, Quantization of Hitchin's Hamiltonians and Hecke eigen-sheaves, preprint, available at http://www.math.uchicago.edu/$\widetilde{\hphantom{m}}$mitya/ langlands.html.
  • A. Bialynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480--497.
  • A. Braverman and M. Finkelberg, Pursuing the double affine Grassmannian II: Convolution, preprint.
  • A. Braverman, M. Finkelberg, and D. Gaitsgory, ``Uhlenbeck spaces via affine Lie algebras'' in The Unity of Mathematics, Progr. Math. 244, Birkhäuser, Boston, 2006, 17--135.
  • A. Braverman, M. Finkelberg, D. Gaitsgory, and I. Mirković, Intersection cohomology of Drinfeld's compactifications, Selecta Math. (N.S.) 8 (2002), 381--418.
  • A. Braverman and D. Gaitsgory, Crystals via the affine Grassmannian, Duke Math. J. 107 (2001), 561--575.
  • A. Braverman and D. Kazhdan, The spherical Hecke algebra for affine Kac-Moody groups I, preprint.
  • R. K. Brylinski, Limits of weight spaces, Lusztig's $q$-analogs, and fiberings of adjoint orbits, J. Amer. Math. Soc. 2 (1989), 517--533.
  • W. Crawley-Boevey, Geometry of the moment map for representations of quivers, Compositio Math. 126 (2001), 257--293.
  • G. Faltings, Algebraic loop groups and moduli spaces of bundles, J. Eur. Math. Soc. (JEMS) 5 (2003), 41--68.
  • S. Fishel, I. Grojnowski, and C. Teleman, The strong Macdonald conjecture and Hodge theory on the loop Grassmannian, Ann. of Math. (2) 168 (2008), 175--220.
  • I. B. Frenkel, ``Representations of affine Lie algebras, Hecke modular forms and Korteweg-de Vries type equations'' in Lie Algebras and Related Topics (New Brunswick, N.J., 1981), Lecture Notes in Math. 933, Springer, Berlin, 1982, 71--110.
  • V. Ginzburg, Perverse sheaves on a loop group and Langlands' duality, preprint,\arxivalg-geom/9511007v4
  • K. Hasegawa, Spin module versions of Weyl's reciprocity theorem for classical Kac-Moody Lie algebras --.-an application to branching rule duality, Publ. Res. Inst. Math. Sci. 25 (1989), 741--828.
  • A. Joseph, G. Letzter, and S. Zelikson, On the Brylinski-Kostant filtration, J. Amer. Math. Soc. 13 (2000), 945--970.
  • M. Kashiwara and T. Tanisaki, Kazhdan-Lusztig conjecture for affine Lie algebras with negative level, Duke Math. J. 77 (1995), 21--62.
  • S. Kato, Spherical functions and a $q$-analogue of Kostant's weight multiplicity formula, Invent. Math. 66 (1982), 461--468.
  • S. Kumar, Kac-Moody Groups, Their Flag Varieties and Representation Theory, Progr. Math. 204, Birkhäuser, Boston, 2002.
  • J. Lepowsky, Calculus of twisted vertex operators, Proc. Nat. Acad. Sci. U.S.A. 82 (1985), 8295--8299.
  • A. Licata, Framed rank $r$ torsion-free sheaves on $\CC{P}^2$ and representations of the affine Lie algebra $\widehat{gl(r)}$, preprint.
  • G. Lusztig, ``Singularities, character formulas, and a $q$-analog of weight multiplicities'' in Analysis and Topology on Singular Spaces, II, III (Luminy, France, 1981), Astérisque 101 --.102, Soc. Math. France, Montrouge, 1983, 208--229.
  • I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2) 166 (2007), 95--143.
  • I. Mirković and M. Vybornov, On quiver varieties and affine Grassmannians of type A, C. R. Math. Acad. Sci. Paris 336 (2003), 207--212.
  • H. Nakajima, ``Geometric construction of representations of affine algebras'' in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 423--438.
  • —, Quiver varieties and branching, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), paper 003.
  • S. Viswanath, Kostka-Foulkes polynomials for symmetrizable Kac-Moody algebras, Sém. Lothar. Combin. 58 (2007/08), Art. B58f.
  • E. Witten, ``Conformal field theory in four and six dimensions'' in Topology, Geometry and Quantum Field Theory, London Math. Soc. Lecture Note Ser. 308, Cambridge Univ. Press, Cambridge, 2004, 405--419.