The t-analogs of q-characters at roots of unity for quantum affine algebras and beyond

Abstract

The q-characters were introduced by Frenkel and Reshetikhin [The q-characters of representations of quantum affine algebras and deformations of W-algebras, in: Recent Developments in Quantum Affine Algebras and Related Topics, in: Contemp. Math., vol. 248, Amer. Math. Soc., Providence, RI, 1999, pp. 163–205] to study finite dimensional representations of the untwisted quantum affine algebra U q( g ̂ ) for q generic. The ε-characters at roots of unity were constructed by Frenkel and Mukhin [The q-characters at roots of unity, Adv. Math. 171 (1) (2002) 139–167] to study finite dimensional representations of various specializations of U q( g ̂ ) at q s =1. In the finite simply laced case Nakajima [ t-analogue of the q-characters of finite dimensional representations of quantum affine algebras, in: Physics and Combinatorics, Proc. Nagoya 2000 Internat. Workshop, World Scientific, Singapore, 2001, pp. 181–212; Quiver varieties and t-analogs of q-characters of quantum affine algebras, Ann. of Math., in press; preprint arXiv: math.QA/0105173] defined deformations of q-characters called q, t-characters for q generic and also at roots of unity. The definition is combinatorial but the proof of the existence uses the geometric theory of quiver varieties which holds only in the simply laced case. In [Algebraic approach to q, t-characters, Adv. Math., in press, preprint arXiv: math.QA/0212257] we proposed an algebraic general (non-necessarily simply laced) new approach to q, t-characters for q generic. In this paper we propose two developments of this approach: we treat the root of unity case and the case of a larger class of generalized Cartan matrices (including finite and affine cases except A 1 (1), A 2 (2)). In particular, we generalize the construction of analogs of Kazhdan–Lusztig polynomials at roots of unity of [H. Nakajima, Quiver varieties and t-analogs of q-characters of quantum affine algebras, Ann. of Math., in press, preprint arXiv: math.QA/0105173] to those cases. We also study properties of various objects used in this article: deformed screening operators at roots of unity, t-deformed polynomial algebras, bicharacters arising from symmetrizable Cartan matrices, deformation of the Frenkel–Mukhin's algorithm.