Affinization of category 𝒪 for quantum groups

Abstract

Let be a simple Lie algebra. We consider the category of those modules over the affine quantum group whose -weights have finite multiplicity and lie in a finite union of cones generated by negative roots. We show that many properties of the category of the finite-dimensional representations naturally extend to the category . In particular, we define the minimal affinizations of parabolic Verma modules. In types ABCFG we classify these minimal affinizations and conjecture a Weyl denominator type formula for their characters.