Heisenberg algebras and rational double affine Hecke algebras

Shan, P.; Vasserot, E.

doi:10.1090/S0894-0347-2012-00738-3

Heisenberg algebras and rational double affine Hecke algebras
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by P. Shan and E. Vasserot

J. Amer. Math. Soc. 25 (2012), 959-1031

DOI: https://doi.org/10.1090/S0894-0347-2012-00738-3

Published electronically: April 23, 2012

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Abstract:

We relate the filtration by the support on the Grothendieck group $[\mathcal {O}]$ of the category $\mathcal {O}$ of cyclotomic rational double affine Hecke algebras to a representation-theoretic grading on $[\mathcal {O}]$, defined using the action of an affine Lie algebra and of a Heisenberg algebra on the Fock space. This implies a recent conjecture of Etingof. The proof uses a categorification of the Heisenberg action, which is new, and a categorification of the affine Lie algebra action, which was introduced by the first author in an earlier paper.

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