We study a family of abelian categories
O
_
c
,
ν
\underline {\mathcal {O}}_{\text { } c,\nu }
depending on complex parameters
c
,
ν
c, \nu
which are interpolations of the category
O
\mathcal {O}
for the rational Cherednik algebra
H
c
(
ν
)
H_c(\nu )
of type
A
A
, where
ν
\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
,
ν
c, \nu
, and prove that a Verma object of infinite length exists in
O
c
,
ν
\mathcal {O}_{\text { } c,\nu }
only if
c
∈
Q
>
0
c \in \mathbb {Q}_{>0}
. We also show that for every
c
∈
Q
>
0
c \in \mathbb {Q}_{>0}
there exists
ν
∈
Q
>
0
\nu \in \mathbb {Q}_{>0}
such that there exists a Verma object of infinite length in
O
c
,
ν
\mathcal {O}_{\text { } c,\nu }
.
The latter result is an example of a degeneration phenomenon which can occur in rational values of
ν
\nu
, as was conjectured by P. Etingof.