Specialization of t-analog q-characters to classical q-characters

Prove that for quantum affine algebras of non-simply-laced type, and for every simple module S(w) in the category O_ξ, the t-analog of the q-character χ_{q,t}(S(w)) specializes at t→1 to the classical q-character χ_q(S(w)). This would confirm that the quantum Grothendieck ring is a t-deformation of the classical Grothendieck ring in these cases.

Background

The paper reviews cluster structures arising from representations of quantum affine algebras and the associated quantum Grothendieck rings defined via t-analogs of q-characters. For simply-laced types, these structures are well understood; for non-simply-laced types, cluster structures exist but certain compatibilities between t-analogs and classical q-characters remain conjectural.

Establishing the specialization χ{q,t}(S(w))|{t→1}=χ_q(S(w)) in non-simply-laced types would align the quantum and classical character theories and strengthen the bridge between representation theory and cluster algebra bases in these settings.