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

Dominik Brennecken

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 Rösler, Koornwinder, and Voit.

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

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 Rösler, Koornwinder, and Voit.

Paper Structure

This paper contains 6 sections, 26 theorems, 84 equations.

Table of Contents

  1. Trigonometric Dunkl theory for integral root systems
  2. Connection to spherical functions of reductive Lie groups
  3. Recurrence relations for the Cherednik operators
  4. Helgason-Johnson theorem for the Cherednik kernel
  5. Non-symmetric limit transition between $\mathrm{BC}_n$ and $\mathrm{A}_n$
  6. Limit transition of the Cherednik kernels

Key Result

Proposition 1.3

For all $\xi \in \mathfrak{a}$ and $f \in C^1(\mathfrak{a})$ we have:

Theorems & Definitions (54)

  • Definition 1.1
  • Remark 1.2
  • Proposition 1.3
  • proof
  • Theorem 1.4
  • proof
  • Proposition 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Theorem 1.8
  • ...and 44 more