Duke Mathematical Journal

Geometric realizations of Wakimoto modules at the critical level

Edward Frenkel and Dennis Gaitsgory

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We study the Wakimoto modules over the affine Kac-Moody algebras at the critical level from the point of view of the equivalences of categories proposed in our previous works, relating categories of representations and certain categories of sheaves. In particular, we explicitly describe geometric realizations of Wakimoto modules as Hecke eigen-D-modules on the affine Grassmannian and as quasi-coherent sheaves on the flag variety of the Langlands dual group

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Duke Math. J., Volume 143, Number 1 (2008), 117-203.

First available in Project Euclid: 23 May 2008

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Zentralblatt MATH identifier

Primary: 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
Secondary: 81R10: Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $W$-algebras and other current algebras and their representations [See also 17B65, 17B67, 22E65, 22E67, 22E70]


Frenkel, Edward; Gaitsgory, Dennis. Geometric realizations of Wakimoto modules at the critical level. Duke Math. J. 143 (2008), no. 1, 117--203. doi:10.1215/00127094-2008-017. https://projecteuclid.org/euclid.dmj/1211574665

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  • S. Arkhipov, R. Bezrukavnikov, and V. Ginzburg, Quantum groups, the loop Grassmannian, and the Springer resolution, J. Amer. Math. Soc. 17 (2004), 595--678.
  • S. Arkhipov, A. Braverman, R. Bezrukavnikov, D. Gaitsgory, and I. Mirković, Modules over the small quantum group and semi-infinite flag manifold, Transform. Groups 10 (2005), 279--362.
  • S. Arkhipov and D. Gaitsgory, Another realization of the category of modules over the small quantum group, Adv. Math. 173 (2003), 114--143.
  • A. Beilinson and V. Drinfeld, Chiral Algebras, Amer. Math. Soc. Colloq. Publ. 51, Amer. Math. Soc., Providence, 2004.
  • —, Quantization of Hitchin's integrable system and Hecke eigensheaves, preprint, available at http://www.math.uchicago.edu/$\sim$mitya/langlands.html
  • B. L. FeĭGin and E. V. Frenkel, A family of representations of affine Lie algebras, Russian Math. Surveys 43, no. 5 (1988), 221--222.
  • —, Affine Kac-Moody algebras and semi-infinite flag manifolds, Comm. Math. Phys. 128 (1990), 161--189.
  • —, ``Affine Kac-Moody algebras at the critical level and Gel'fand-Dikiĭ algebras'' in Infinite Analysis, Part A (Kyoto, 1991), Adv. Ser. Math. Phys. 16, World Sci., River Edge, N.J., 1992, 197--215.
  • E. Frenkel, Wakimoto modules, opers and the center at the critical level, Adv. Math. 195 (2005), 297--404.
  • E. Frenkel and D. Ben-Zvi, Vertex Algebras and Algebraic Curves, 2nd ed., Math. Surveys Monogr. 88, Amer. Math. Soc., Providence, 2004.
  • E. Frenkel and D. Gaitsgory, D-modules on the affine Grassmannian and representations of affine Kac-Moody algebras, Duke Math. J. 125 (2004), 279--327.
  • —, Fusion and convolution: Applications to affine Kac-Moody algebras at the critical level, Pure Appl. Math. Q. 2 (2006), 1255--1312.
  • —, ``Local geometric Langlands correspondence and affine Kac-Moody algebras'' in Algebraic Geometry and Number Theory, Progr. Math. 253, Birkhäuser, Boston, 2006, 69--260.
  • —, Localization of, $\hg$-modules on the affine Grassmannian, preprint,\arxivmath/0512562v1[math.RT]
  • D. Gaitsgory, The notion of category over an algebraic stack,\arxivmath/0507192v1[math.AG]
  • I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2) 166 (2007), 95--143.
  • M. Wakimoto, Fock representations of affine Lie algebra $A_1^(1)$, Comm. Math. Phys. 104 (1986), 605--609.