Extended TQ-relations in the Grothendieck ring K0(O)

Establish the extended TQ-relations in K0(O) for all w ∈ W: for any finite-dimensional simple U_q(ĝ)-module V, after replacing each variable Y_{i,a} in the q-character X_q(V) by the specified product [w(ω_i)]·[L(Y_{−w(ω_i), a q^{−1}})]/[L(Y_{w(ω_i), a q})], the resulting expression, once denominators are cleared, equals [V].

Background

These relations generalize the Baxter-type TQ-relations proved previously for the longest Weyl group element w0. They encode identities among classes of simple modules in the category O derived from substitutions in q-characters using extremal l-weights Y_{w(w_i),a}.

The conjecture is proven for w = w0 and for simple reflections; the general case would yield a Weyl-group–indexed family of spectral descriptions for transfer-matrices via Baxter operators.