Summary
We prove that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to the algebra of functions on the space of monodromy-free opers on the disc with regular singularity and residue determined by the highest weight of the Weyl module. This result may be used to test the local geometric Langlands correspondence proposed in our earlier work.
This is a preview of subscription content, log in via an institution to check access.
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
T. Arakawa, Representation theory of \(\mathcal{W}\) -algebras, Preprint arXiv: math/0506056.
A.Beilinson and V.Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves, available at http://www.math.uchicago.edu/arinkin/langlands/
A. Beilinson and V. Drinfeld, Opers, Preprint arXiv:math.AG/0501398.
V. Drinfeld and V. Sokolov, Lie algebras and KdV type equations, J. Sov. Math. 30 (1985) 1975–2036.
D. Eisenbud and E. Frenkel, Appendix to M. Mustaţă, Jet schemes of locally complete intersection canonical singularities, Inv. Math. 145 (2001) 397–424.
B. Feigin and E. Frenkel, Affine Kac–Moody algebras at the critical level and Gelfand–Dikii algebras, in Infinite Analysis, eds. A. Tsuchiya, T. Eguchi,M. Jimbo, Adv. Ser. in Math. Phys. 16, 197–215, Singapore: World Scientific, 1992.
E. Frenkel, Affine algebras, Langlands duality and Bethe Ansatz, in Proceedings of the International Congress of Mathematical Physics, Paris, 1994, ed. D. Iagolnitzer, pp. 606–642, International Press, 1995; arXiv:q-alg/9506003.
E. Frenkel, Wakimoto modules, opers and the center at the critical level, Adv. Math. 195 (2005) 297–404; arXiv:math.QA/0210029.
E. Frenkel and D. Ben-Zvi, Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs 88, Second Edition, AMS, 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, Local geometric Langlands correspondence and affine Kac–Moody algebras, in Algebraic Geometry and Number Theory, Progress in Math. 253, pp. 69–260, Birkhäuser Boston, 2006.
E. Frenkel and D. Gaitsgory, Fusion and convolution: applications to affine Kac–Moody algebras at the critical level, Pure and Applied Math. Quart. 2, (2006) 1255–1312.
E. Frenkel, V. Kac, and M. Wakimoto, Characters and fusion rules for \(\mathcal{W}\) –algebras via quantized Drinfeld-Sokolov reduction, Comm. Math. Phys. 147 (1992) 295–328.
V.G. Kac, Infinite-dimensional Lie Algebras, 3rd Edition, Cambridge University Press, 1990.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Frenkel, E., Gaitsgory, D. (2010). Weyl Modules and Opers without Monodromy. In: Ceyhan, Ö., Manin, Y.I., Marcolli, M. (eds) Arithmetic and Geometry Around Quantization. Progress in Mathematics, vol 279. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4831-2_5
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4831-2_5
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4830-5
Online ISBN: 978-0-8176-4831-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)