Polynomiality of renormalized X-series operators

Establish that for every w ∈ W and i ∈ I, the renormalized X-series operator X^N_{w(w_i)}(z), obtained by dividing the X-series X_{w(w_i)}(z) = T_w(X_i(z)) by its eigenvalue on the l-weight corresponding to T_w(m), is a polynomial in z when acting on any simple finite-dimensional U_q(ĝ)-module L(m).

Background

The X-series are formal power series in the Drinfeld–Cartan elements defined via the braid group action T_w on the fundamental series X_i(z). The authors prove rationality for all w and show polynomiality in special cases (w = e and w = w0), and for simple reflections.

This conjecture strengthens Conjecture 6.8 by asserting polynomiality of the operator itself, not just its eigenvalues, and it implies the extremal monomial property via an equivalence with Conjecture 4.4.