Dual extended TQ-relations in the Grothendieck ring K0(O*)

Establish the dual extended TQ-relations in K0(O*): for any w ∈ W and finite-dimensional simple U_q(ĝ)-module V, after replacing each Y_{i,a} in X_q(V) by the ratio [-w(ω_i)]·[L'(Y^{-1}_{w(ω_i), a q^{−1}})]/[L'(Y^{-1}_{w(ω_i), a q})], the resulting expression, once denominators are cleared, equals [T_{q^{−2 r_v h_v}}(V^*)].

Background

This is the dual version of the extended TQ-relations for the category O*, shown in the paper to be equivalent to Conjecture 7.12. It is tailored to the transfer-matrices built from lowest l-weight modules L'(·), aligning with the generalized Baxter operators considered in the dual category.

Proving these relations would yield spectral identities for transfer-matrices and facilitate polynomial descriptions of spectra via dual Baxter operators.