Polynomiality of generalized Baxter operators

Establish that for every w ∈ W and i ∈ I, the renormalized generalized Baxter transfer-matrix (f_{i,m}(z))^{-1} t^{w}_{w_i}(z, u) acts as a polynomial in z on any simple finite-dimensional U_q(ĝ)-module L(m).

Background

The operators t^{w}_{w_i}(z,u) are transfer-matrices associated with simple modules L'(Ω(Y_{w(w_i),1})) in the dual category O*, and in the limit of the twist parameters they degenerate to the X-series X_{w(w_i)}(z). Polynomiality is known for w = e, and an analogue for the longest element connects to results in [Z].

This conjecture extends the Baxter polynomiality phenomenon (previously proved for w = w0) to a Weyl-group–indexed family and underpins multiple alternative spectral parametrizations for XXZ-type quantum integrable models.