  The q-characters were introduced by Frenkel and mReshetikhin to study finite
dimensional representations of the untwisted quantum affine algebra for q
generic. The $\epsilon$-characters at roots of unity were constructed by
Frenkel and Mukhin to study finite dimensional representations of various
specializations of the quantum affine algebra at $q^s=1$. In the finite simply
laced case Nakajima 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 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 treat the root of unity case. Moreover we
construct q-characters and q,t-characters for a large class of generalized
Cartan matrices (including finite and affine cases except $A_1^{(1)}$,
$A_2^{(2)}$) by extending our algebraic approach. In particular we generalize
the construction of Nakajima's Kazhdan-Lusztig polynomials at roots of unity 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.

Quantum Algebra