The Heat Equation on a Compact Lie Group

Abstract

Recently there has been much work related to Macdonald's $\eta$-function identities. In the present paper the aim is to give another proof of these identities using analytical methods. This is done by using the heat equation to obtain Kostant's form of the identities. The basic idea of the proof is to look at subgroups of the Lie group which are isomorphic to the group $SU(2)$. When this has been done the problem has essentially been reduced to that for the group $SU(2)$, which is a classical result.