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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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This presumably finds its origins in the analogous averaging trick for finite groups, which was certainly known to the founding fathers of representation theory of finite groups such as Burnside, Frobenius and Schur. –  Geoff Robinson Jul 24 '12 at 14:15
    
The references at en.wikipedia.org/wiki/Unitarian_trick might tell you. –  Qiaochu Yuan Jul 24 '12 at 14:16
    
@QiaochuYuan: At Wikipedia, there is an explicit reference to a paper by Hurwitz but none to Weyl, though his name is mentioned. –  Arnold Neumaier Jul 24 '12 at 14:29
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@Arnold: yes, I mean consulting the references listed there might tell you the answer, not that the references themselves are the answer. –  Qiaochu Yuan Jul 24 '12 at 14:30
    
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). –  Marc van Leeuwen Jul 25 '12 at 22:14

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up vote 8 down vote accepted

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". 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. –  t.b. Jul 25 '12 at 2:56

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