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.
The relationship between G and K that drives this connection 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.
The "trick" is stated in a number of ways in contemporary mathematics.
Then Gan, the complex points of G considered as a Lie group, has a compact subgroup K that is Zariski-dense.
For any such group G and maximal compact subgroup K, and V a complex vector space of finite dimension which is a G-module, its G-submodules and K-submodules are the same.
[3] In the Encyclopedia of Mathematics, the formulation is The classical compact Lie groups ... have the same complex linear representations and the same invariant subspaces in tensor spaces as their complex envelopes [...].
Therefore, results of the theory of linear representations obtained for the classical complex Lie groups can be carried over to the corresponding compact groups and vice versa.
[4]In terms of Tannakian formalism, Claude Chevalley interpreted Tannaka duality starting from a compact Lie group K as constructing the "complex envelope" G as the dual reductive algebraic group Tn(K) over the complex numbers.
[5] Veeravalli S. Varadarajan wrote of the "unitarian trick" as "the canonical correspondence between compact and complex semisimple complex groups discovered by Weyl", noting the "closely related duality theories of Chevalley and Tannaka", and later developments that followed on quantum groups.
Weyl extended Schur's method to complex semisimple Lie algebras by showing they had a compact real form.
[7] The complete reducibility of finite-dimensional linear representations of compact groups, or connected semisimple Lie groups and complex semisimple Lie algebras goes sometimes under the name of Weyl's theorem.
It was proved by Weyl a few years before "universal cover" had a formal definition.