Radial part calculations for affine sl2 and the Heun KZB-heat equation

TL;DR

This paper computes the radial part of the Casimir element for affine sl2, linking it to the Inozemtsev Hamiltonian and KZB-heat equation, and explores associated spherical functions and theta functions.

Contribution

It explicitly determines the radial part of the Casimir element for affine sl2 and connects it to known integrable systems and special functions, extending previous work.

Findings

Identification of the radial part with a blend of Inozemtsev Hamiltonian and KZB-heat equation

Construction of symmetric theta functions from zonal spherical functions

Discussion of convergence properties of the functions

Abstract

In the present paper we determine the radial part of the Casimir element for the Lie algebra affine sl2 with respect to the Chevalley involution. The resulting operator is identified with a blend of the Inozemtsev Hamiltonian and the KZB-heat equation in dimension one. Moreover, it is shown how the corresponding zonal spherical functions give rise to symmetric theta functions and convergence is discussed. The paper takes guidance from previous work by Etingof and Kirillov on the diagonal case.