## A geometric proof of the Feigin-Frenkel theorem

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- by Sam Raskin
- Represent. Theory
**16**(2012), 489-512 - DOI: https://doi.org/10.1090/S1088-4165-2012-00417-4
- Published electronically: September 20, 2012
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Previous version: Original version posted September 20, 2012

## Abstract:

We reprove the theorem of Feigin and Frenkel relating the center of the critical level enveloping algebra of the Kac-Moody algebra for a semisimple Lie algebra to opers (which are certain de Rham local systems with extra structure) for the Langlands dual group. Our proof incorporates a construction of Beilinson and Drinfeld relating the Feigin-Frenkel isomorphism to (more classical) Langlands duality through the geometric Satake theorem.## References

- A. Beilinson,
*Remarks on topological algebras*, Mosc. Math. J.**8**(2008), no. 1, 1–20, 183 (English, with English and Russian summaries). MR**2422264**, DOI 10.17323/1609-4514-2008-8-1-1-20 - A. Beilinson and V. Drinfeld, “Quantization of Hitchin’s integrable system and Hecke eigensheaves.” Available at: http://math.uchicago.edu/\textasciitilde mitya/langlands/hitchin/BD-hitchin.pdf
- Alexander Beilinson and Vladimir Drinfeld,
*Chiral algebras*, American Mathematical Society Colloquium Publications, vol. 51, American Mathematical Society, Providence, RI, 2004. MR**2058353**, DOI 10.1090/coll/051 - Edward Frenkel and David Ben-Zvi,
*Vertex algebras and algebraic curves*, 2nd ed., Mathematical Surveys and Monographs, vol. 88, American Mathematical Society, Providence, RI, 2004. MR**2082709**, DOI 10.1090/surv/088 - V. G. Drinfel′d,
*Commutative subrings of certain noncommutative rings*, Funkcional. Anal. i Priložen.**11**(1977), no. 1, 11–14, 96 (Russian). MR**0476732** - V. G. Drinfel′d and V. V. Sokolov,
*Lie algebras and equations of Korteweg-de Vries type*, Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 81–180 (Russian). MR**760998** - Mircea Mustaţă,
*Jet schemes of locally complete intersection canonical singularities*, Invent. Math.**145**(2001), no. 3, 397–424. With an appendix by David Eisenbud and Edward Frenkel. MR**1856396**, DOI 10.1007/s002220100152 - Edward Frenkel,
*Wakimoto modules, opers and the center at the critical level*, Adv. Math.**195**(2005), no. 2, 297–404. MR**2146349**, DOI 10.1016/j.aim.2004.08.002 - Boris Feigin and Edward Frenkel,
*Affine Kac-Moody algebras at the critical level and Gel′fand-Dikiĭ algebras*, Infinite analysis, Part A, B (Kyoto, 1991) Adv. Ser. Math. Phys., vol. 16, World Sci. Publ., River Edge, NJ, 1992, pp. 197–215. MR**1187549**, DOI 10.1142/s0217751x92003781 - Edward Frenkel and Dennis Gaitsgory,
*$D$-modules on the affine Grassmannian and representations of affine Kac-Moody algebras*, Duke Math. J.**125**(2004), no. 2, 279–327. MR**2096675**, DOI 10.1215/S0012-7094-04-12524-2 - Edward Frenkel and Dennis Gaitsgory,
*Local geometric Langlands correspondence and affine Kac-Moody algebras*, Algebraic geometry and number theory, Progr. Math., vol. 253, Birkhäuser Boston, Boston, MA, 2006, pp. 69–260. MR**2263193**, DOI 10.1007/978-0-8176-4532-8_{3} - Edward Frenkel and Dennis Gaitsgory,
*Fusion and convolution: applications to affine Kac-Moody algebras at the critical level*, Pure Appl. Math. Q.**2**(2006), no. 4, Special Issue: In honor of Robert D. MacPherson., 1255–1312. MR**2282421**, DOI 10.4310/PAMQ.2006.v2.n4.a14 - Edward Frenkel and Dennis Gaitsgory,
*Geometric realizations of Wakimoto modules at the critical level*, Duke Math. J.**143**(2008), no. 1, 117–203. MR**2414746**, DOI 10.1215/00127094-2008-017 - Susanna Fishel, Ian Grojnowski, and Constantin Teleman,
*The strong Macdonald conjecture and Hodge theory on the loop Grassmannian*, Ann. of Math. (2)**168**(2008), no. 1, 175–220. MR**2415401**, DOI 10.4007/annals.2008.168.175 - Edward Frenkel and Constantin Teleman,
*Self-extensions of Verma modules and differential forms on opers*, Compos. Math.**142**(2006), no. 2, 477–500. MR**2218907**, DOI 10.1112/S0010437X05001958 - Robert Gilmer and William Heinzer,
*The Noetherian property for quotient rings of infinite polynomial rings*, Proc. Amer. Math. Soc.**76**(1979), no. 1, 1–7. MR**534377**, DOI 10.1090/S0002-9939-1979-0534377-2 - Bertram Kostant,
*On Whittaker vectors and representation theory*, Invent. Math.**48**(1978), no. 2, 101–184. MR**507800**, DOI 10.1007/BF01390249 - Jacob Lurie,
*Higher topos theory*, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. MR**2522659**, DOI 10.1515/9781400830558 - J. Lurie, “Higher algebra.” Available at: http://math.harvard.edu/\textasciitilde lurie/papers/HigherAlgebra.pdf
- I. Mirković and K. Vilonen,
*Geometric Langlands duality and representations of algebraic groups over commutative rings*, Ann. of Math. (2)**166**(2007), no. 1, 95–143. MR**2342692**, DOI 10.4007/annals.2007.166.95

## Bibliographic Information

**Sam Raskin**- Affiliation: Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138
- Email: sraskin@math.harvard.edu
- Received by editor(s): June 12, 2011
- Received by editor(s) in revised form: August 21, 2011, and January 3, 2012
- Published electronically: September 20, 2012
- © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**16**(2012), 489-512 - MSC (2010): Primary 17B65, 81R10, 14D24
- DOI: https://doi.org/10.1090/S1088-4165-2012-00417-4
- MathSciNet review: 2972556