Hernandez conjecture in full generality

Establish that for every complex finite-dimensional simple Lie algebra g and every dominant monomial m in the Frenkel–Reshetikhin Y-variables, the specialization at t=1 of the simple (q,t)-character equals the q-character, i.e., ev_{t=1} χ_{q,t}(L(m)) = χ_q(L(m)) for the simple U_q(Lg)-module L(m) whose highest ℓ-weight corresponds to m.

Background

For simply-laced types ADE, Nakajima constructed (q,t)-characters via quiver varieties and proved that their specialization at t=1 recovers Frenkel–Reshetikhin q-characters. Hernandez later gave an algebraic Kazhdan–Lusztig–type algorithm that constructs simple (q,t)-characters for arbitrary types, leading to a natural conjecture that the specialization identity holds uniformly.

Recent advances verified the conjecture for type B and for all simples in certain core subcategories, and the present work proves it for all classical types A, B, C, and D. However, the conjecture remains unresolved beyond these cases, notably for exceptional types, motivating the full generality statement.