Extremal monomial property of q-characters for all Weyl group elements

Prove that for every Weyl group element w ∈ W and every simple finite-dimensional U_q(ĝ)-module L(m) with highest monomial m, the q-character satisfies X_q(L(m)) ∈ T_w(m) · Z[(A^w_{j,b})^{-1}]_{j∈I,b∈C×}, where T_w is Chari’s braid group action and A^w_{j,b} := T_w(A_{j,b}) are the w-twisted root monomials.

Background

This conjecture generalizes the “highest monomial property” (the case w = e) proved by Frenkel–Mukhin to an ‘extremal’ version for all w ∈ W using Chari’s braid group action. It predicts that the entire q-character is generated from the extremal monomial T_w(m) by nonpositive powers of the w-twisted root monomials.

The conjecture is known for w = e and for the longest element w0, and is proved in the paper for simple reflections. A weaker inclusion allowing positive powers (Theorem 4.10) is established for all w. The conjecture is equivalent to the polynomiality of associated X-series eigenvalues (Conjecture 6.8).