Traces of intertwiners for quantum affine algebras and difference equations (after Etingof-Schiffmann-Varchenko)

TL;DR

This paper refines and proves results on traces of intertwiners for quantum affine algebras, demonstrating that certain normalized traces satisfy multiple q-difference equations and exhibit symmetry properties.

Contribution

It provides complete proofs and modifications for existing results, showing these traces solve four key q-difference systems and revealing their symmetry, extending prior work by Etingof-Schiffmann-Varchenko.

Findings

Normalized traces solve Macdonald-Ruijsenaars and dual systems

Traces satisfy q-KZB and dual q-KZB equations

Trace functions exhibit symmetry properties

Abstract

We modify and give complete proofs for the results of Etingof-Schiffmann-Varchenko 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 for $U_q(\hat{\mathfrak{g}})$ 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 recent work of the author.