On representations of rational Cherednik algebras of complex rank

We study a family of abelian categories O _c,ν depending on complex parameters c, ν which are interpolations of the category O for the rational Cherednik algebra H_c(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 ν Q<₀. We also show that for every c ε Q<₀ there exists ν ε 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.