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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?

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    I'm not sure the question correctly describes the unitary trick. I would think that the essential point is to draw conclusions about certain _noncompact_ groups by making use of a certain compact subgroup (for instance the subgroup pf unitary matrices in the case of matrix groups).2012-07-25

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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.

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    Weyl himself called the argument *The unitarian trick* (not the unitarity trick) in his book "[The classical groups](http://books.google.com/books?id=zmzKSP2xTtYC&pg=PA265)". Quoting from p.176 of the 1945 edition: *We therefore take refuge in what might be called the unitarian trick: each group is replaced by the subgroup of those elements that are unitary transformations.*2012-07-25