A new integral formula for Heckman–Opdam hypergeometric functions

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 and Kirillov Jr. in [10]. 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 [4] of Borodin and Gorin on the β-Jacobi corners ensemble.