On representations of rational Cherednik algebras of complex rank

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.