Monomials of q and q, t-characters for non simply-laced quantum affinizations

Abstract

Nakajima [23][24] introduced the morphism of q, t-characters for finite dimensional representation of simply-laced quantum affine algebras: it is a t-deformation of the Frenkel-Reshetikhin’s morphism of q-characters (sum of monomials in infinite variables). In [15] we generalized the construction of q, t-characters for non simply-laced quantum affine algebras. First in this paper we prove a conjecture of [15]: the monomials of q and q, t-characters of standard representations are the same in non simply-laced cases (the simply-laced cases were treated in [24]) and the coefficients are non negative. In particular those q, t-characters can be considered as t-deformations of q-characters. In the proof we show that for quantum affine algebras of type A, B, C and quantum toroidal algebras of type A(1) the l-weight spaces of fundamental representations are of dimension 1. Eventually we show and use a generalization of a result of [13][10][22]: for general quantum affinizations we prove that the l-weights of a l-highest weight simple module are lower than the highest l-weight in the sense of monomials.