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Classification of Harish-Chandra Modules for Current Algebras


Author: Michael Lau
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
Published electronically: October 5, 2017
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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.


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Michael Lau
Affiliation: Département de mathématiques et de statistique, Université Laval Québec, QC, Canada G1V 0A6
Email: Michael.Lau@mat.ulaval.ca

DOI: https://doi.org/10.1090/proc/13834
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
Article copyright: © Copyright 2017 American Mathematical Society

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