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Representations of infinitesimal Cherednik algebras


Authors: Fengning Ding and Alexander Tsymbaliuk
Journal: Represent. Theory 17 (2013), 557-583
MSC (2010): Primary 17B10
DOI: https://doi.org/10.1090/S1088-4165-2013-00443-0
Published electronically: November 5, 2013
MathSciNet review: 3123740
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Abstract: Infinitesimal Cherednik algebras are continuous analogues of rational Cherednik algebras, and in the case of $ \mathfrak{gl}_n$, are deformations of universal enveloping algebras of the Lie algebras $ \mathfrak{sl}_{n+1}$. In the first half of this paper, we compute the determinant of the Shapovalov form, enabling us to classify all irreducible finite dimensional representations of $ H_\zeta (\mathfrak{gl}_n)$. In the second half, we investigate Poisson-analogues of the infinitesimal Cherednik algebras and generalize various results to $ H_\zeta (\mathfrak{sp}_{2n})$, including Kostant's theorem.


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Additional Information

Fengning Ding
Affiliation: Phillips Academy, 180 Main St., Andover, Massachusetts 01810
Address at time of publication: Department of Mathematics, Harvard College, Cambridge, Massachusetts 02138
Email: fding@college.harvard.edu

Alexander Tsymbaliuk
Affiliation: Independent University of Moscow, 11 Bol’shoy Vlas’evskiy per., Moscow 119002, Russia
Address at time of publication: Department of Mathematics, MIT, 77 Massachusetts Ave., Cambridge, Massachusetts 02139
Email: sasha{\textunderscore}ts@mit.edu

DOI: https://doi.org/10.1090/S1088-4165-2013-00443-0
Received by editor(s): October 21, 2012
Received by editor(s) in revised form: February 26, 2013, March 30, 2013, and July 31, 2013
Published electronically: November 5, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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