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On representations of rational Cherednik algebras of complex rank


Author: Inna Entova Aizenbud
Journal: Represent. Theory 18 (2014), 361-407
MSC (2010): Primary 16S99, 18D10
DOI: https://doi.org/10.1090/S1088-4165-2014-00459-X
Published electronically: November 24, 2014
MathSciNet review: 3280664
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Abstract: We study a family of abelian categories $ \underline {\mathcal {O}}_{\text { } c,\nu }$ depending on complex parameters $ c, \nu $ which are interpolations of the category $ \mathcal {O}$ for the rational Cherednik algebra $ H_c(\nu )$ of type $ A$, where $ \nu $ is a positive integer. We define the notion of a Verma object in such a category (a natural analogue of the notion of Verma module).

We give some necessary conditions and some sufficient conditions for the existence of a non-trivial morphism between two such Verma objects. We also compute the character of the irreducible quotient of a Verma object for sufficiently generic values of parameters $ c, \nu $, and prove that a Verma object of infinite length exists in $ \mathcal {O}_{\text { } c,\nu }$ only if $ c \in \mathbb{Q}_{<0}$. We also show that for every $ c \in \mathbb{Q}_{<0}$ there exists $ \nu \in \mathbb{Q}_{<0}$ such that there exists a Verma object of infinite length in $ \mathcal {O}_{\text { } c,\nu }$.

The latter result is an example of a degeneration phenomenon which can occur in rational values of $ \nu $, as was conjectured by P. Etingof.


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Additional Information

Inna Entova Aizenbud
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: inna.entova@gmail.com

DOI: https://doi.org/10.1090/S1088-4165-2014-00459-X
Keywords: Deligne categories, rational Cherednik algebra
Received by editor(s): March 17, 2014
Received by editor(s) in revised form: June 14, 2014, and September 17, 2014
Published electronically: November 24, 2014
Article copyright: © Copyright 2014 American Mathematical Society