Definition 8

We denote by $\widehat{G}$ the set of classes of irreducible unitary representations of G in a complex Hilbert space. One says that $\widehat{G}$ is the unitary dual of G.

For every irreducible unitary representation $\pi$ of G in a complex Hilbert space, one therefore has cl($\pi$ )$\in \widehat{G}$.