Representations of Affine Lie Algebras, Parabolic Differential Equations and Lam E Functions

We consider correlation functions for the Wess-Zumino-Witten model on the torus with the insertion of a Cartan element; mathematically this means that we consider the function of the form F = Tr((1 (z 1) : : : n (z n)q ?@ e h) where i are intertwiners between Verma modules and evaluation modules over an aane Lie algebra ^ g, @ is the grading operator in a Verma module and h is in the Cartan subalgebra of g. We derive a system of diierential equations satissed by such a function. In particular, the calculation of q @ @q F yields a parabolic second order PDE closely related to the heat equation on the compact Lie group corresponding to g (cf. Ber]). We consider in detail the case n = 1, g = sl 2. In this case we get the following diierential equation (q = e i): ?2i(K + 2) @ @ + @ 2 @x 2 F = (m(m + 1)}(x + 2) + c)F, which for K = ?2 (critical level) becomes Lam e equation. For the case m 2 Z we derive integral formulas for F and nd their asymptotics as K ! ?2, thus recovering classical Lam e functions.