Representations of infinitesimal Cherednik algebras

Ding, Fengning; Tsymbaliuk, Alexander

doi:10.1090/S1088-4165-2013-00443-0

Representations of infinitesimal Cherednik algebras
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by Fengning Ding and Alexander Tsymbaliuk

Represent. Theory 17 (2013), 557-583

DOI: https://doi.org/10.1090/S1088-4165-2013-00443-0

Published electronically: November 5, 2013

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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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