M ar 2 00 5 CHARACTERS OF FUNDAMENTAL REPRESENTATIONS OF QUANTUM

We give closed formulae for the q–characters of the fundamental representations of the quantum loop algebra in terms of a family of partitions satisfying some simple properties. We also give the multiplicities of the eigenvalues of the imaginary subalgebra in terms of these partitions. Introduction In this paper we study the q–characters of the fundamental finite–dimensional representations of the quantum loop algebraUq associated to a classical simple Lie algebra. The notion of q–characters defined in [6] is analogous to the usual notion of a character of a finite–dimensional representation of a simple Lie algebra. These characters and their generalizations have been studied extensively [5], [7], [9] using combinatorial and geometric methods. A more representation theoretic approach was developed in [3]. In particular, that paper approached the problem of studying whether the q–characters admitted a Weyl group invariance which was analogous to the invariance of characters of finite–dimensional representations of simple Lie algebras. In the quantum case, it is reasonable to expect that the Weyl group be replaced by the braid group, and it was shown in [3] that the q–characters of the fundamental representations, are in a suitable sense invariant under the braid group action. It was also shown that the q–character of such representations could then be calculated in a certain inductive way. In this paper, we use that inductive method to give closed formulas for the q–characters of all the fundamental representations of the quantum loop algebras of a classical simple Lie algebra. To describe the results a bit further, recall that the quantum loop algebra admits a commutative subalgebra Uq(0) corresponding to the imaginary root vectors. Any finite–dimensional representation V of the quantum loop algebra, breaks up as a direct sum of generalized eigenspaces for the action of Uq(0). These are called the l–weight spaces and the eigenvalues corresponding to the non–zero eigenspaces are called the l–weights of the representation. The l–weights lie in a free abelian multiplicative group Pq. Let Z[Pq] be the integral group ring over Pq and for ̟ ∈ Pq, let V̟ be the corresponding eigenspace of V . The element of Z[Pq] defined by, chl(V ) = ∑