## On representations of rational Cherednik algebras of complex rank

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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
- 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
- © Copyright 2014 American Mathematical Society
- Journal: Represent. Theory
**18**(2014), 361-407 - MSC (2010): Primary 16S99, and, 18D10
- DOI: https://doi.org/10.1090/S1088-4165-2014-00459-X
- MathSciNet review: 3280664