Boundedness of the Cherednik kernel and its limit transition from type BC to type A

TL;DR

This paper introduces a Cherednik kernel for root systems, characterizes its boundedness, and establishes a limit transition from type BC to type A, generalizing classical theorems and previous hypergeometric function results.

Contribution

It generalizes the Helgason-Johnson theorem to Cherednik kernels and proves a new limit transition between types A and BC for these kernels.

Findings

Characterization of bounded Cherednik kernels for integral root systems.

Limit transition from type BC to type A Cherednik kernel.

Generalization of classical theorems to a broader setting.

Abstract

We introduce a Cherednik kernel and a hypergeometric function for integral root systems and prove their relation to spherical functions associated with Riemannian symmetric spaces of reductive Lie groups. Furthermore, we characterize the spectral parameters for which the Cherednik kernel is a bounded function. In the case of a crystallographic root system, this characterization was proven by Narayanan, Pasquale and Pusti for the hypergeometric function. This result generalizes the Helgason- Johnson theorem from 1969, which characterizes the bounded spherical functions of a Riemannian symmetric space. The characterization for the Cherednik kernel is based on recurrence relations for the associated Cherednik operators under the dual affine Weyl group going back to Sahi. These recurrence relations are also used to prove a limit transition between the Cherednik kernel of type A and of type B, which generalizes an already known result for the associated hypergeometric functions by R\"osler, Koornwinder, and Voit.