A new integral formula for Heckman-Opdam hypergeometric functions

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.

30 pages. v4: corrected sign convention in definition of CM Hamiltonians and Dunkl operators; corrected computations of adjoints in Sections 4 and 5; main results are unchanged

Cited by in corpus (6)

• Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the $S$-transform
• Matrix models for multilevel Heckman-Opdam and multivariate Bessel measures
• Regularized limits of Stokes matrices, isomonodromy deformation and crystal basis
• Sonine formulas and intertwining operators in Dunkl theory
• Relations between generalised Wishart matrices, the Muttalib--Borodin model and matrix spherical functions
• Quantum $S{L}_2$, Infinite curvature and Pitman's 2M-X theorem