Abstract
Let \( \mathfrak{g} \) be a simple Lie algebra over ℂ and G a connected algebraic group with Lie algebra \( \mathfrak{g} \). The affine Kac-Moody algebra \( \hat {\mathfrak{g}} \) is the universal central extension of the formal loop agebra \( \mathfrak{g} \)((t)). Representations of \( \hat {\mathfrak{g}} \) have a parameter, an invariant bilinear form on \( \mathfrak{g} \), which is called the level. Representations corresponding to the bilinear form which is equal to minus one half of the Killing form are called representations of critical level. Such representations can be realized in spaces of global sections of twisted D-modules on the quotient of the loop group G((t)) by its “open compact” subgroup K, such as G[[t]] or the Iwahori subgroup I.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
S. Arkhipov, A new construction of the semi-infinite BGG resolution, q-alg/9605043, 1996.
A. Arkhipov and R. Bezrukavnikov, Perverse sheaves on affine flags and Langlands dual group, math.RT/0201073, 2002.
S. Arkhipov, R. Bezrukavnikov, V. Ginzburg, Quantum groups, the loop Grassmannian, and the Springer resolution, J. Amer. Math. Soc., 17 (2004), 595–678.
S. Arkhipov and D. Gaitsgory, Differential operators on the loop group via chiral algebras, Internat. Math. Res. Notices, 2002-4 (2002), 165–210.
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 J. Bernstein, Localisation de g-modules, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 15–18.
A. Beilinson and V. Drinfeld, Chiral Algebras, American Mathematical Society Colloquium Publications, Vol. 51, American Mathematical Society, Providence, RI, 2004.
A. Beilinson, Tensor products of topological vector spaces, addendum to [CHA].
A. Beilinson and V. Drinfeld, Quantization of Hitchin’s Integrable System and Hecke Eigensheaves, available online from http://www.math.uchicago.edu/~arinkin/langlands/.
A. Beilinson and V. Drinfeld, Opers, math.AG/0501398, 2005.
A. Beilinson and V. Ginzburg, Wall-crossing functors and D-modules, Representation Theory, 3 (1999), 1–31.
J. Bernstein, Trace in categories, in Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989), Progress in Mathematics, Vol. 92, Birkhäuser Boston, Cambridge, MA, 1990, 417–423.
J. Bernstein and V. Luntz, Equivariant Sheaves and Functors, Lecture Notes in Mathematics, Vol. 1578, Springer-Verlag, Berlin, 1994.
R. Bezrukavnikov, unpublished manuscript.
V. Drinfeld, DG quotients of DG categories, J. Algebra, 272 (2004), 643–691.
V. Drinfeld, Infinite-dimensional vector bundles in algebraic geometry, in P. Etinghof, V. Retakh, and I. Singer, eds., The Unity of Mathematics, Birkhäuser Boston, Cambridge, MA, 2005, 263–304.
V. Drinfeld and V. Sokolov, Lie algebras and KdV type equations, J. Soviet Math., 30 (1985), 1975–2036.
D. Eisenbud and E. Frenkel, Appendix to M. MustaţĂ, “Jet schemes of locally complete intersection canonical singularities,” Invent. Math., 145-3 (2001), 397–424.
B. Feigin, The semi-infinite cohomology of Kac-Moody and Virasoro Lie algebras, Russian Math. Survey, 39-2 (1984), 155–156.
B. Feigin and E. Frenkel, A family of representations of affine Lie algebras, Russian Math. Survey, 43-5 (1988), 221–222.
B. Feigin and E. Frenkel, Affine Kac-Moody algebras and semi-infinite flag manifolds, Comm. Math. Phys., 128 (1990), 161–189.
B. Feigin and E. Frenkel, Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras, in A. Tsuchiya, T. Eguchi, and M. Jimbo, eds., Infinite Analysis, Advanced Series in Mathematical Physics, Vol. 16, World Scientific, Singapore, 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., Mathematical Surveys and Monographs, Vol. 88, American Mathematical Society, Providence, RI, 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.
E. Frenkel, D. Gaitsgory, and K. Vilonen, Whittaker patterns in the geometry of moduli spaces of bundles on curves, Ann. Math., 153 (2001), 699–748.
E. Frenkel and K. Teleman, Self-extensions of Verma modules and differential forms on opers, Compositio Math., 142 (2006), 477–500.
D. Gaitsgory, Construction of central elements in the affine Hecke algebra via nearby cycles, Invent. Math., 144 (2001), 253–280.
D. Gaitsgory, The notion of category over an algebraic stack, math.AG/0507192, 2005.
D. Gaitsgory, Notes on Bezrukavnikov’s theory, in preparation.
V. Gorbounov, F. Malikov, and V. Schechtman, On chiral differential operators over homogeneous spaces, Internat. J. Math. Math. Sci., 26 (2001), 83–106.
V. Kac and D. Kazhdan, Structure of representations with highest weight of infinite-dimensional Lie algebras, Adv. Math., 34 (1979), 97–108.
B. Kostant, Lie group representations on polynomial rings, Amer. J. Math., 85 (1963), 327–402.
R. P. Langlands, Problems in the theory of automorphic forms, in Lectures in Modern Analysis and Applications III, Lecture Notes in Mathematics, Vol. 170, Springer-Verlag, Berlin, New York, Heidelberg, 1970, 18–61.
G. Laumon, Transformation de Fourier, constantes d’équations fonctionelles et conjecture de Weil, Publ. I.H.E.S., 65 (1987), 131–210.
K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Mathematical Society Lecture Note Series, Vol. 124, Cambridge University Press, Cambridge, UK, 1987.
I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, math.RT/0401222, 2004.
D. Vogan, Local Langlands correspondence, in Representation Theory of Groups and Algebras, Contemporary Mathematics, Vol. 145, American Mathematical Society, Providence, RI, 1993, 305–379.
A. Voronov, Semi-infinite induction and Wakimoto modules, Amer. J. Math., 121 (1999), 1079–1094.
M. Wakimoto, Fock representations of affine Lie algebra A (1)1 , Comm. Math. Phys., 104 (1986), 605–609.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to Vladimir Drinfeld on his 50th birthday.
Rights and permissions
Copyright information
© 2006 Birkhäuser Boston
About this chapter
Cite this chapter
Frenkel, E., Gaitsgory, D. (2006). Local geometric Langlands correspondence and affine Kac-Moody algebras. In: Ginzburg, V. (eds) Algebraic Geometry and Number Theory. Progress in Mathematics, vol 253. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4532-8_3
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4532-8_3
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4471-0
Online ISBN: 978-0-8176-4532-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)