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In the present paper, we determine the radial part of the Casimir element for the affine Lie algebra   $\widehat {\mathfrak {sl}}_2$   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. Communicated by Prof. Eric Opdam

Radial Part Calculations for $\widehat {\mathfrak {sl}}_2$ and the Heun-KZB Heat Equation