AFFINIZATION OF CATEGORY O FOR QUANTUM GROUPS

Let g be a simple Lie algebra. We consider the category $\hat{\mathrm{O}}$ of those modules over the affine quantum group ${\mathrm{U}}_{\mathrm{q}}\left(\hat{\mathrm{g}}\right)$ whose ${\mathrm{U}}_{\mathrm{q}}\left(\mathrm{g}\right)$-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 $\hat{\mathrm{O}}$. 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.