For any compact Lie group $K$, the loop group ${\cal L} \equiv C(S^1; K)$ possesses both a natural gradient operator and a natural probability measure, $\mu$. The associated “Laplacian” in $L^2(\cal L,\mu )$ has been the subject of investigations which are aimed at developing a Hodge–deRham theorem over $\cal L$. The harmonic functions over $\cal L$ have recently been described. The techniques of proof have led to a collection of unitary isomorphisms between functions over $K$, holomorphic functions over the complexification of $K$ and a Hilbert completion of the universal enveloping algebra.