Classification of Harish-Chandra Modules for Current Algebras
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Abstract:
For any reductive Lie algebra $\mathfrak {g}$ and commutative, associative, unital algebra $S$, we give a complete classification of the simple weight modules of $\mathfrak {g}\otimes S$ with finite weight multiplicities. In particular, any such module is parabolically induced from a simple admissible module for a Levi subalgebra. Conversely, all modules obtained in this way have finite weight multiplicities. These modules are isomorphic to tensor products of evaluation modules at distinct maximal ideals of $S$. Our results also classify simple Harish-Chandra modules up to isomorphism for all central extensions of current algebras.References
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Additional Information
- Michael Lau
- Affiliation: Département de mathématiques et de statistique, Université Laval, Québec, QC, Canada G1V 0A6
- MR Author ID: 760608
- Email: Michael.Lau@mat.ulaval.ca
- Received by editor(s): June 29, 2016
- Received by editor(s) in revised form: May 10, 2017
- Published electronically: October 5, 2017
- Additional Notes: Funding from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
- Communicated by: Kailash C. Misra
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 1015-1029
- MSC (2010): Primary 17B10; Secondary 17B65, 17B67, 17B22
- DOI: https://doi.org/10.1090/proc/13834
- MathSciNet review: 3750215