Proposition 3

Suppose that $G = G_1\times G_2$ where $G_1$ and $G_2$ are compact groups. The mapping $b$ from $\widehat{G}_1\times \widehat{G}_2$ into $\widehat{G}$ which to $(\pi_1, \pi_2)$ associates the class of $\pi_1\boxtimes \pi_2$ is a bijection.

By cor. 6 of V, p. 389, the mapping $b$ is well defined.

Let $\pi_1$ and $\pi_2$ be elements of $\widehat{G}$. Let $\pi \in \widehat{G}$. The $\pi$-isotypical component of the restriction $\varpi$ of $\pi_1\boxtimes \pi_2$ to $G_1\times \{e\}$ is equal to $\varpi$ if $\pi_1=\pi$ and is zero in the contrary case (lemma 8 of V, p. 384). Thus, the unitary representation $\pi_1\boxtimes \pi_2$ determines $\pi_1$ up to isomorphism; analogously, it determines $\pi_2$, which proves that $b$ is injective.

Let us prove that $b$ is surjective. Let $\pi$ be an irreducible unitary representation of $G_1\times G_2$ in a Hilbert space E. It is finite-dimensional (prop. 2). Let $\pi_1$ be an irreducible unitary representation of $G_1$ in a Hilbert space $E_1$ such that the restriction of $\pi$ to $G_1\times \{e\}$ contains a subrepresentation isomorphic to $\pi_1$ (lemma 3 of V, p. 379). Let F = Hom$_{G_1}(\pi_1, \pi )$. It is a nonzero finite-dimensional vector space. For $h\in G_2$ and $u\in F$, let $\varrho (h)(u)$ be the linear mapping from $E_1$ into E defined by $x\mapsto \pi (e, h)(u(x))$. Since $G_1\times \{e\}$ and $\{e\} \times G_2$ commute in G, the mapping $\varrho (h)(u)$ belongs to the space F. The mapping $\varrho$ is a linear representation of $G_2$ in the space F; it is continuous. Since F is not reduced to 0, there exists an irreducible subrepresentation $\pi_2$ of $\varrho$ (lemma 3 of V, p. 379). Let $E_2$ be the space of $\pi_2$. The linear mapping $v: E_1\otimes E_2\rightarrow E$ such that $x\otimes u\mapsto u(x)$ for $x\in E_1$ and $u\in E_2$ is then a nonzero G-morphism of $\pi_1\boxtimes \pi_2$ into $\pi$; since the representations $\pi$ and $\pi_1\boxtimes \pi_2$ are irreducible and finite-dimensional, the morphism $v$ is an isomorphism (lemma 2 of V, p. 378).

This proposition is to be compared with theorem 1 of A, VIII, p. 208.