Schur orthogonality relations and invariant sesquilinear forms

Donley, Robert, Jr.

doi:10.1090/S0002-9939-01-06227-X

Schur orthogonality relations and invariant sesquilinear forms
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by Robert W. Donley Jr.

Proc. Amer. Math. Soc. 130 (2002), 1211-1219

DOI: https://doi.org/10.1090/S0002-9939-01-06227-X

Published electronically: August 29, 2001

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Abstract:

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 $V$.

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