A local Peter-Weyl theorem

An invariant inner product on the Lie algebra of a compact connected Lie group extends to a Hermitian inner product on the Lie algebra of the complexified Lie group . The Laplace-Beltrami operator, , on induced by the Hermitian inner product determines, for each number , a Green's function by means of the identity . The Hilbert space of holomorphic functions on which are square integrable with respect to is shown to be finite dimensional. It is spanned by the holomorphic extensions of the matrix elements of those irreducible representations of whose Casimir operator is appropriately related to .