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Compatibility of the Feigin-Frenkel Isomorphism and the Harish-Chandra Isomorphism for jet algebras


Author: Masoud Kamgarpour
Journal: Trans. Amer. Math. Soc. 368 (2016), 2019-2038
MSC (2010): Primary 17B67, 17B69, 22E50, 20G25
DOI: https://doi.org/10.1090/S0002-9947-2014-06419-2
Published electronically: October 3, 2014
MathSciNet review: 3449232
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Abstract: Let $ \mathfrak{g}$ be a simple finite-dimensional complex Lie algebra with a Cartan subalgebra $ \mathfrak{h}$ and Weyl group $ W$. Let $ \mathfrak{g}_n$ denote the Lie algebra of $ n$-jets on $ \mathfrak{g}$. A theorem of Raïs and Tauvel and Geoffriau identifies the centre of the category of $ \mathfrak{g}_n$-modules with the algebra of functions on the variety of $ n$-jets on the affine space $ \mathfrak{h}^*/W$. On the other hand, a theorem of Feigin and Frenkel identifies the centre of the category of critical level smooth modules of the corresponding affine Kac-Moody algebra with the algebra of functions on the ind-scheme of opers for the Langlands dual group. We prove that these two isomorphisms are compatible by defining the higher residue of opers with irregular singularities. We also define generalized Verma and Wakimoto modules and relate them by a nontrivial morphism.


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Additional Information

Masoud Kamgarpour
Affiliation: School of Mathematics and Physics, The University of Queensland, St. Lucia, Queensland 4072, Australia
Email: masoud@uq.edu.au

DOI: https://doi.org/10.1090/S0002-9947-2014-06419-2
Keywords: Irregular opers, residue, jet algebras, Verma modules, Wakimoto modules
Received by editor(s): August 21, 2013
Received by editor(s) in revised form: February 7, 2014
Published electronically: October 3, 2014
Additional Notes: The author was supported by the Australian Research Council Discovery Early Career Research Award
Article copyright: © Copyright 2014 American Mathematical Society