## Abstract

We study a family of abelian categories O c,ν depending on com-

plex parameters c, ν which are interpolations of the category O for the rational

Cherednik algebra Hc(ν) of type A, where ν 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, ν, and prove that a Verma object

of infinite length exists in O c,ν only if c ∈ Q<0. We also show that for every

c ∈ Q<0 there exists ν ∈ Q<0 such that there exists a Verma object of infinite

length in O c,ν .

The latter result is an example of a degeneration phenomenon which can

occur in rational values of ν, as was conjectured by P. Etingof.

plex parameters c, ν which are interpolations of the category O for the rational

Cherednik algebra Hc(ν) of type A, where ν 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, ν, and prove that a Verma object

of infinite length exists in O c,ν only if c ∈ Q<0. We also show that for every

c ∈ Q<0 there exists ν ∈ Q<0 such that there exists a Verma object of infinite

length in O c,ν .

The latter result is an example of a degeneration phenomenon which can

occur in rational values of ν, as was conjectured by P. Etingof.

Original language | English GB |
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Pages (from-to) | 361-407 |

Number of pages | 47 |

Journal | Representation Theory |

Volume | 18 |

Issue number | 1 |

DOIs | |

State | Published - 2014 |

Externally published | Yes |

## Keywords

- Deligne categories
- rational Cherednik algebra