Hermite Functions on Compact Lie Groups, II

The usual formula for Hermite polynomials on Rd is extended to a compact Lie group G, yielding an isometry of L2(G, p1), where p1 is the heat kernel measure at time one, with a natural completion of the universal enveloping algebra of G. The existence of such an isometry was first established by L. Gross; here we present a simplified proof as well as the explicit form of the isometry.