MINIMAL AFFINIZATIONS OF REPRESENTATIONS OF QUANTUM GROUPS - THE IRREGULAR CASE

Let g be a finite-dimensional complex simple Lie algebra and U-q(g) th e associated quantum group (q is a nonzero complex number which we ass ume is transcendental). If V is a finite-dimensional irreducible repre sentation of U-q(g), an affinization of V is an irreducible representa tion (V) over cap of the quantum affine algebra U-q((q) over cap) whic h contains V with multiplicity one and is such that all other irreduci ble U-q(g)-components of (V) over cap have highest weight strictly sma ller than the highest weight lambda of V. There is a natural partial o rder 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 t o U-q(g)-isomorphism; (ii) if g is of type D or E and lambda is not or thogonal to the triple node of the Dynkin diagram of g, there are eith er one or three minimal affinizations (depending on lambda). In this p aper, we show, in contrast to the regular case, that if U-q(g) is of t ype D-4 and lambda is orthogonal to the triple node, the number of min imal affinizations has no upper bound independent of lambda. As a by-p roduct of our methods, we disprove a conjecture according to which, if g is of type A(n), every affinization is isomorphic to a tensor produ ct 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).