Remark

Let $\pi$ (resp. $\varrho$ ) be an irreducible unitary representation of a topological group G (resp. H) in a Hilbert space E (resp. F). Let $\chi$ (resp. $\eta$ ) denote the central character of $\pi$ (resp. $\varrho$ ). The central character of the representation $\overline{\pi}$ is $\overline{\chi}$, and the central character of the irreducible unitary representation $\pi \boxtimes \varrho$ of $G\times H$ (cor. 6) is the character $\chi \boxtimes \eta : (g, h)\mapsto \chi (g)\eta (h)$ of $G\times H$.