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We modify and give complete proofs for the results of Etingof–Schiffmann–Varchenko in [ESV02] on traces of intertwiners of untwisted quantum affine algebras in the opposite coproduct and the standard grading. More precisely, we show that certain normalized generalized traces \( {F}^{V_1,\dots, {V}_n}\left({z}_1,\dots, {z}_n;\uplambda, \omega, \mu, k\right) \) for \( {U}_q\left(\widehat{\mathfrak{g}}\right) \) solve four commuting systems of q-difference equations: the Macdonald–Ruijsenaars, dual Macdonald–Ruijsenaars, q-KZB, and dual q-KZB equations. In addition, we show a symmetry property for these renormalized trace functions. Our modifications are motivated by their appearance in the recent work [Sun16].

TRACES OF INTERTWINERS FOR QUANTUM AFFINE ALGEBRAS AND DIFFERENCE EQUATIONS (AFTER ETINGOF–SCHIFFMANN–VARCHENKO)