  This paper is a sequel to arXiv:1401.6321. We define and study representation
categories based on Deligne categories Rep(GL_t), Rep(O_t), Rep(Sp_2t), where t
is any (non-integer) complex number. Namely, we define complex rank analogs of
the parabolic category O and the representation categories of real reductive
Lie groups and supergroups, affine Lie algebras, and Yangians. We develop a
framework and language for studying these categories, prove basic results about
them, and outline a number of directions of further research.
