Schur orthogonality relations and invariant sesquilinear forms (Q2781273)

Let \(G\) be a connected compact (real) Lie group and \((\pi,V^\pi), (\pi',V^{\pi'})\) irreducible (strongly continuous) unitary representations of \(G\). A substantial connection between the representation theory of \(G\) and the complex Hilbert space \(L^2(G)\) is supplied by the Schur orthogonality relations NEWLINE\[NEWLINE\int_G\langle\pi(g)u,v\rangle \overline{\langle\pi'(g)u',v'\rangle} dg = \begin{cases}\frac{1}{\text{d}_\pi} \langle u,u'\rangle\overline{\langle v,v'\rangle} &\text{if }\pi \cong \pi'\\ 0 &\text{otherwise},\end{cases}NEWLINE\]NEWLINE where \(u, v\) are in \(V^\pi\), \(u', v'\) are in \(V^{\pi'}\), \(dg\) is the normalized Haar measure, and \(\text{d}_\pi\) is the dimension of \(V^\pi\). In this paper, the author first recasts the Schur orthogonality relations without using the invariant measure and then generalizes these relations to all finite-dimensional representations of a connected semisimple Lie group with finite center. Since a similar theory would be desirable for a separable locally compact group \(G\), a general framework is established in the case of irreducible continuous unitary representations \((\pi,V)\) of \(G\), where \(V\) is a separable complex Hilbert space, by identifying the linear span of the matrix coefficients of \(\pi\) with a dense subspace of the vector space of all Hilbert-Schmidt endomorphisms of \(V\).