Convolution algebras for Heckman-Opdam polynomials derived from compact Grassmannians

TL;DR

This paper develops new convolution algebras (hypergroups) related to Heckman-Opdam polynomials of type BC, derived from compact Grassmannians, providing explicit product formulas and extending previous non-compact case results.

Contribution

It introduces three continuous classes of positive convolution algebras for root systems of type BC from compact Grassmannians, linking them to explicit product formulas for Heckman-Opdam polynomials.

Findings

Derived positive convolution algebras (hypergroups) for type BC root systems.

Established explicit positive product formulas for Heckman-Opdam polynomials.

Connected convolution structures to spherical functions of compact Grassmannians.

Abstract

We study convolution algebras associated with Heckman-Opdam polynomials. For root systems of type BC we derive three continuous classes of positive convolution algebras (hypergroups) by interpolating the double coset convolution structures of compact Grassmannians U/K with fixed rank over the real, complex or quaternionic numbers. These convolution algebras are linked to explicit positive product formulas for Heckman-Opdam polynomials of type BC, which occur for certain discrete multiplicities as the spherical functions of U/K. These results complement those of a recent paper by the second author for the non-compact case.