Classical counterpart of non-linear highest weight quantum modules

Determine the classical counterpart, in the representation theory of the universal enveloping algebra U(g), of an irreducible highest weight U_q-module L_q(Λ) when the highest weight Λ ∈ T_k is not a linear weight (i.e., Λ ≠ q^λ for any λ ∈ 𝔥_ℚ^*). Ascertain whether such a counterpart exists and, if it does, construct or characterize it explicitly; if it does not exist, establish precise criteria for non-existence.

Background

In the generic quantum case, weights Λ ∈ T_k that are linear (Λ = q^λ for λ in 𝔥_ℚ^*) lead to simple modules whose behavior upon specialization q → 1 aligns with the classical representation theory of U(g); in particular, the corresponding primitive ideals and Gelfand–Kirillov dimensions coincide with those in the classical setting by results of Joseph–Letzter and subsequent work.

However, the paper notes that for non-linear weights Λ ∈ T_k, there is no clear established correspondence to classical U(g)-modules, raising the question of existence and nature of such a counterpart. Resolving this would clarify the bridge between quantum and classical representation theories beyond the linear-weight regime and determine whether a systematic classical analogue can be defined for these quantum modules.