On representations of rational Cherednik algebras of complex rank

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.