v 2 2 0 O ct 1 99 3 REPRESENTATIONS OF AFFINE LIE ALGEBRAS , PARABOLIC DIFFERENTIAL EQUATIONS AND LAMÉ 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(Φ 1 (z 1). .. Φ n (z n)q −∂ e h) where Φ i are intertwiners between Verma modules and evaluation modules over an affine Lie algebrâ g, ∂ is the grading operator in a Verma module and h is in the Cartan subalgebra of g. We derive a system of differential equations satisfied 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 differential equation (q = e πiτ): −2πi(K + 2) ∂ ∂τ + ∂ 2 ∂x 2 F = (m(m + 1)℘(x + τ 2) + c)F , which for K = −2 (critical level) becomes Lamé equation. For the case m ∈ Z we derive integral formulas for F and find their asymptotics as K → −2, thus recovering classical Lamé functions.