A local Peter-Weyl theorem

An Ad K invariant inner product on the Lie algebra of a compact connected L ie group K extends to a Hermitian inner product on the Lie algebra of the c omplexified Lie group K-c. The Laplace-Beltrami operator, Delta, on K-c ind uced by the Hermitian inner product determines, for each number a > 0, a Gr een's function r(a) by means of the identity (a(2) -Delta/4)(-1) = r(a*). T he Hilbert space of holomorphic functions on Kc which are square integrable with respect to r(a)(x)dx is shown to be finite dimensional. It is spanned by the holomorphic extensions of the matrix elements of those irreducible representations of K whose Casimir operator is appropriately related to a.