Minimal affinizations of representations of quantum groups: The irregular case

Let g be a finite-dimensional complex simple Lie algebra and U-q(g) the associated quantum group (q is a nonzero complex number which we assume is transcendental). If V is a finite-dimensional irreducible representation of U-q(g), an affinization of V is an irreducible representation (V) over cap of the quantum affine algebra U-q((q) over cap) which contains V with multiplicity one and is such that all other irreducible U-q(g)-components of (V) over cap have highest weight strictly smaller than the highest weight lambda of V. There is a natural partial order on the set of U-q(g)-isomorphism classes of affinizations, and we look for the minimal one(s). In earlier papers, we showed that (i) if g is of type A, B, C, F or G, the minimal affinization is unique up to U-q(g)-isomorphism; (ii) if g is of type D or E and lambda is not orthogonal to the triple node of the Dynkin diagram of g, there are either one or three minimal affinizations (depending on lambda). In this paper, we show, in contrast to the regular case, that if U-q(g) is of type D-4 and lambda is orthogonal to the triple node, the number of minimal affinizations has no upper bound independent of lambda. As a by-product of our methods, we disprove a conjecture according to which, if g is of type A(n), every affinization is isomorphic to a tensor product of representations of U-q((g) over cap) which are irreducible under U-q(g) (in an earlier paper, we proved this conjecture when n = 1).