A new integral formula for Heckman-Opdam hypergeometric functions

TL;DR

This paper introduces a new integral formula for Heckman-Opdam hypergeometric functions, connecting representation theory, integrable systems, and special functions through Harish-Chandra type integrals and limits of Macdonald polynomials.

Contribution

It provides a novel Harish-Chandra type integral representation for Heckman-Opdam functions using dressing orbits and limits of Macdonald polynomials, linking multiple areas in mathematical physics.

Findings

Derived Harish-Chandra type formulas for multivariate Bessel and Heckman-Opdam functions.

Connected integral representations to Gelfand-Tsetlin polytopes and beta-Jacobi ensembles.

Provided new proofs and interpretations of existing integral formulas in representation theory.

Abstract

We provide Harish-Chandra type formulas for the multivariate Bessel functions and Heckman-Opdam hypergeometric functions as representation-valued integrals over dressing orbits. Our expression is the quasi-classical limit of the realization of Macdonald polynomials as traces of intertwiners of quantum groups given by Etingof-Kirillov Jr. Integration over the Liouville tori of the Gelfand-Tsetlin integrable system and adjunction for higher Calogero-Moser Hamiltonians recovers and gives a new proof of the integral realization over Gelfand-Tsetlin polytopes which appeared in the recent work of Borodin-Gorin on the beta-Jacobi corners ensemble.