Weyl's unitarity trick creates from an irreducible representation of a compact group a unitary representation by averaging with a Haar measure.
Does anyone know a reference to the paper (or book, with page number) where Weyl introduced his unitarity trick?
The first instance I'm aware of is in Hermann Weyl, Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. I, Math. Z. 23 (1925), pages 271-309. (Freely available on GDZ)
See in particular §5, "Der Satz von der vollen Reduzibilität", starting on page 288.