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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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On representations of rational Cherednik algebras of complex rank
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by Inna Entova Aizenbud PDF
Represent. Theory 18 (2014), 361-407 Request permission

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.

References
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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