Polynomiality of eigenvalues of renormalized X-series

Establish that for every w ∈ W and i ∈ I, every eigenvalue of the renormalized X-series X^N_{w(w_i)}(z) acting on any simple finite-dimensional U_q(ĝ)-module L(m) is a polynomial in z.

Background

The authors construct X-series X_{w(w_i)}(z) as Tw-transforms of fundamental series and normalize them by removing the eigenvalue on the extremal l-weight. They prove eigenvalue rationality in general and polynomiality in specific cases (w = e, w = w0, and simple reflections).

They also show this conjecture is equivalent to the extremal monomial property (Conjecture 4.4), making it central to connecting q-characters and spectral theory of XXZ-type models.