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Monomials of \(q\) and \(q,t\)-characters for non simply-laced quantum affinizations. (English) Zbl 1098.17010
Author’s abstract: H. Nakajima [Proceedings of the Nagoya 2000 2nd international workshop, Nagoya, Japan, 2000, 196–219 (2001; Zbl 1011.17013), (*) Ann. Math. (2) 160, No. 3, 1057–1097 (2005; Zbl 1140.17015), see also arXiv:math.QA/0105173] 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 [D. Hernandez, Adv. Math. 187, No. 1, 1–52 (2004; Zbl 1098.17006)] we generalized the construction of \(q\), \(t\)-characters for non simply-laced quantum affine algebras. First in this paper we prove a conjecture of [Adv. Math., loc. cit.]: 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 [(*) loc. cit.]) and the coefficients are nonnegative. 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 [E. Frenkel and N. Reshetikhin, Contemp. Math. 248, 163–205 (1999; Zbl 0973.17015), E. Frenkel and E. Mukhin, Commun. Math. Phys. 216, 23–57 (2001; Zbl 1051.17013), H. Nakajima, J. Am. Math. Soc. 14, No. 1, 145–238 (2001; Zbl 0981.17016)]: 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.

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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