The heat equation on compact Lie group

Abstract

McKean and Singer [9] posed the problem of the existence of an analogue of the Poisson's summation formula for manifolds other than flat tori. Y. Colin de Verdiere [3] gave an answer to it in the case of a 2-dimensional compact Riemannian manifold with negative sectional curvature. The purpose of this paper is to determine the Minakshisundaram's expansion (Theorem 3) related to the heat equation and to give an analogue of the Poisson's summation formula of the fundamental solution of this equation on a simply-connected compact semi-simple Lie group (Theorem 2). The author expresses his gratitude to Prof. K. Okamoto who suggested him to study the Poisson's formula when he obtained Theorem 3.