Lemma 2

Let $\varrho_1$ be a nonzero unitary representation of G isomorphic to a subrepresentation of $\varrho$. There exists a measure $\nu \in M$ such that $\varrho_1(\check{\nu}*\nu )$ is a nonzero hermitian compact endomorphism of the space of $\varrho_1$. In particular, there exists a nonzero real number $\lambda$ such that the eigensubspace of $\varrho_1(\check{\nu}*\nu )$ corresponding to $\lambda$ is nonzero.

One may suppose that $\varrho_1$ is a subrepresentation of $\varrho$. Let $E_1$ be its space. For every measure $\nu \in M$, the endomorphism $\varrho_1(\nu )$ is deduced from $\varrho (\nu )$ by passing to subspaces. The hypothesis and prop. 3 of III, p. 5 therefore imply that $\varrho_1(\nu )$ is compact.

Since $\mathfrak{F}$ converges to the measure $\varepsilon_e$ in $\mathscr{M}_c(G)$ endowed with the topology of compact convergence and since the space $E_1$ is nonzero, there exists $\nu \in M$ such that $v=\varrho_1(\nu )$ is a nonzero endomorphism of $E_1$ (cf. No.$^o2$ of V, p. 400). The endomorphism $u=\varrho_1(\check{\nu}*\nu )$ is equal to $v^*\circ v$ (lemma 1 of V, p. 401); it is therefore nonzero (EVT, V, p. 39, prop. 2), hermitian and compact, since $v$ is compact.

Since $u$ is hermitian and nonzero, its spectrum is not reduced to zero (example 1 of I, p. 110). Finally, since $u$ is compact, every $\lambda \in$ Sp($u$)$-\{0\}$ is an eigenvalue of $u$ (prop. 2 of III, p. 83).

Let us now prove the proposition.

We shall use prop. 7 of V, p. 391 to establish that $\varrho$ is semisimple. Let $\varrho_1$ be a nonzero subrepresentation of $\varrho$, and $E_1\subset E$ its space; we must prove that $\varrho_1$ contains an irreducible subrepresentation.

Let $\nu \in M$ be such that $u=\varrho_1(\check{\nu}*\nu )$ is a nonzero compact hermitian endomorphism of the space of $\varrho_1$, and let $\lambda$ be nonzero such that the eigensubspace F of $u$ relative to $\lambda$ is nonzero (lemma 2). The space F is of finite dimension (prop. 5 of III, p. 90). There exists a subrepresentation $\varrho_2$ of $\varrho_1$ on a subspace $E_2$ of $E_1$ such that $E_2\cap F$ is nonzero and of minimal dimension. Let $x$ be a nonzero element of $E_2\cap F$, and let $E_3$ be the subrepresentation of $\varrho_1$ generated by $x$. Then $E_3\subset E_2$, whence $E_3\cap F = E_2\cap F$ by minimality of the dimension of $E_2\cap F$.

Let us prove that the representation $E_3$ is irreducible. Let $E_4\subset E_3$ be a closed subspace stable under G. We have $E_2\cap F = (E_4\cap F)\oplus (E^{\circ}_4\cap F)$, where $E^{\circ}_4$ denotes the orthogonal of $E_4$ in $E_3$ (in fact, if $y\in E_2\cap F$, let $y_4\in E_4$ be its orthogonal projection on $E_4$; since $E_4$ and $E^{\circ}_4$ are stable under $u$, the vector $u(y_4)$ is the orthogonal projection of $u(y) =\lambda y$ on $E_4$, whence $u(y_4) =\lambda y_4$, which means that $y_4\in E_4\cap F$, and the assertion follows). The minimality of the dimension of $E_2\cap F$ then implies that either $E_4\cap F = E_2\cap F$, or $E^{\circ}_4\cap F = E_2\cap F$. In the first case, we have $x\in E_4$, whence $E_4= E_3$, and in the second, we have $x\in E^{\circ}_4$, whence $E^{\circ}_4= E_3$ and $E_4=\{0\}$. The representation $E_3$ is therefore irreducible.

It follows from Prop. 7 of V, p. 391 that the representation $\varrho$ is semisimple.

Let $\pi$ be an irreducible unitary representation of G whose multiplicity in $\varrho$ is nonzero; let $E_{\pi}$ be its space. There exist a measure $\nu \in M$ and a nonzero real number $\lambda$ such that $u=\pi (\check{\nu}*\nu )$ is a nonzero compact hermitian endomorphism of $E_{\pi}$ and that the proper subspace of $u$ corresponding to $\lambda$ is nonzero (Lemma 2). Let F be the proper subspace of $v=\varrho (\check{\nu}*\nu )$ corresponding to $\lambda$. For every subrepresentation $E_1$ of E isomorphic to $\pi$, the endomorphism of $E_1$ deduced from $v$ by passing to the subspaces is identified with $u$, hence the intersection of $E_1$ and F is nonzero. Thus, the multiplicity of $\pi$ in $\varrho$ is less than the dimension of the space F, which is finite (Prop. 5 of III, p. 90).