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

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.