Affinization of category O for quantum groups

Let g be a simple Lie algebra. We consider the category ˆO of those modules over the affine quantum group Uq(bg) whose Uq(g)-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 ˆO . In particular, we develop the theory of q-characters and 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.