Connection coefficients for basic Harish-Chandra series

TL;DR

This paper explicitly computes connection coefficients for basic Harish-Chandra series, linking them to elliptic solutions of root system analogs of quantum equations, with applications in hypergeometric functions and integrable systems.

Contribution

It provides explicit formulas for connection coefficients in terms of theta functions and interprets them as transition functions for solutions of quantum root system equations.

Findings

Connection coefficients expressed via Jacobi theta functions.

Explicit elliptic solutions to dynamical Yang-Baxter and reflection equations.

Applications to hypergeometric functions and integrable difference operators.

Abstract

Basic Harish-Chandra series are asymptotically free meromorphic solutions of the system of basic hypergeometric difference equations associated to root systems. The associated connection coefficients are explicitly computed in terms of Jacobi theta functions. We interpret the connection coefficients as the transition functions for asymptotically free meromorphic solutions of Cherednik's root system analogs of the quantum Knizhnik-Zamolodchikov equations. They thus give rise to explicit elliptic solutions of root system analogs of dynamical Yang-Baxter and reflection equations. Applications to quantum c-functions, basic hypergeometric functions, reflectionless difference operators and multivariable Baker-Akhiezer functions are discussed.