Minimal affinizations of representations of quantum groups: the irregular case

Let be a finite-dimensional complex simple Lie algebra and U_q() the associated quantum group (q is a nonzero complex number which we assume is transcendental). IfV is a finitedimensional irreducible representation of U_q(), an affinization ofV is an irreducible representationVV of the quantum affine algebra U_q() which containsV with multiplicity one and is such that all other irreducible U_q()-components ofV have highest weight strictly smaller than the highest weight ofV. There is a natural partial order on the set of U_q() classes of affinizations, and we look for the minimal one(s). In earlier papers, we showed that (i) if is of typeA, B, C, F orG, the minimal affinization is unique up to U_q()-isomorphism; (ii) if is of typeD orE and is not orthogonal to the triple node of the Dynkin diagram of, there are either one or three minimal affinizations (depending on ). In this paper, we show, in contrast to the regular case, that if U_q() is of typeD₄ and is orthogonal to the triple node, the number of minimal affinizations has no upper bound independent of .As a by-product of our methods, we disprove a conjecture according to which, if is of typeA_n,every affinization is isomorphic to a tensor product of representations of U_q() which are irreducible under U_q() (in an earlier paper, we proved this conjecture whenn=1).Both authors were partially supported by the NSF, DMS-9207701.