Schur orthogonality relations and invariant sesquilinear forms

Important connections between the representation theory of a compact group $G$ and $L^{2}(G)$ are summarized by the Schur orthogonality relations. The first part of this work is to generalize these relations to all finite-dimensional representations of a connected semisimple Lie group $G.$ The second part establishes a general framework in the case of unitary representations $(\pi , V)$ of a separable locally compact group. The key step is to identify the matrix coefficient space with a dense subset of the Hilbert-Schmidt endomorphisms on $V$.