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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Compatibility of the Feigin-Frenkel Isomorphism and the Harish-Chandra Isomorphism for jet algebras
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by Masoud Kamgarpour PDF
Trans. Amer. Math. Soc. 368 (2016), 2019-2038 Request permission

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
  • 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
  • © Copyright 2014 American Mathematical Society
  • 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
  • MathSciNet review: 3449232