Unitarian trick

In mathematics, the unitarian trick (occasionally unitary trick) is a device in the representation theory of Lie groups, introduced by Hermann Weyl. It applies to show that the representation theory of some group "G" is in a qualitative way controlled by that of some other compact group "K". An important example is that in which "G" is the complex general linear group, and "K" the unitary group acting on vectors of the same size. From the fact that the representations of "K" are completely reducible, the same is concluded for those of "G", at least in finite dimensions.

The relationship between "G" and "K" that makes this connection function is traditionally expressed in the terms that the Lie algebra of "K" is a real form of that of "G". In the theory of algebraic groups, the relationship can also be put that "K" is a dense subset of "G", for the Zariski topology.