Unitarian trick

Device in the representation theory of Lie groups

In mathematics, the unitarian trick (or unitary trick) is a device in the representation theory of Lie groups, introduced by Adolf Hurwitz (1897) for the special linear group and by Hermann Weyl for general semisimple groups. It applies to show that the representation theory of some complex Lie group G is in a qualitative way controlled by that of some compact real Lie group K, and the latter representation theory is easier. An important example is that in which G is the complex general linear group GLn(C), and K the unitary group U(n) acting on vectors of the same size. From the fact that the representations of K are completely reducible, the same is concluded for the complex-analytic representations of G, at least in finite dimensions.

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