Representation theory in complex rank, II

Abstract

We define and study representation categories based on Deligne categories Rep(GL[subscript t]),Rep(O[subscript t]),Rep(Sp₂t), 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. Keywords: Deligne category; Symmetric tensor category; Complex rank