A local Peter-Weyl theorem

An $Ad\, K$ invariant inner product on the Lie algebra of a compact connected Lie group $K$ extends to a Hermitian inner product on the Lie algebra of the complexified Lie group $K_{c}$. The Laplace-Beltrami operator, $\Delta$, on $K_{c}$ induced by the Hermitian inner product determines, for each number $a>0$, a Green's function $r_{a}$ by means of the identity $(a^{2} -\Delta /4 )^{-1} = r_{a} *$. The Hilbert space of holomorphic functions on $K_{c}$ 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 $a$.