  Let g be a simple Lie algebra. We consider the category O-hat of those
modules over the affine quantum group Uq(g-hat) 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 O-hat. 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.
