Example

Suppose that G is unimodular, for example that G is a real semisimple Lie group. Let $\Gamma \subset G$ be a discrete subgroup such that the quotient $X = \Gamma \backslash G$ is compact. The group G acts on the right by multiplication on X. Let $\beta$ denote the counting measure on Γ and put $\widetilde{\mu}=\mu/\beta$ (INT, VII, p. 44, § 2, No.$^o2$, Def. 1); it is a bounded G-invariant measure on X.

For every $f\in \mathscr{L}^2(X,\widetilde{\mu})$ and every $g\in$ G, one defines the function $\varrho (g)f\in \mathscr{L}^2(X,\widetilde{\mu})$ by $\varrho (g)f(x) =f(x\cdot g)$. The mapping $\varrho (g)$ is a continuous linear mapping, which induces by passing to quotients a unitary mapping on $L^2(X,\widetilde{\mu})$, again denoted by $\varrho (g)$. The mapping $\varrho$ is a unitary representation of G in $L^2(X,\widetilde{\mu})$ (INT, VII, p. 135, § 2, No.$^o5$, Prop. 8).

Let $\varphi \in \mathscr{K}(G)$. For all compact subsets $L_1$ and $L_2$ of G, the intersection T of Γ and the compact set $L_1$Supp($\varphi$ )$L^{-1}_2$ is finite. For every $(g, h)\in L_1\times L_2$, the series $\sum_{\gamma\in\Gamma}\varphi (g^{-1}\gamma h)$ coincides with the sum

finite $\sum_{\gamma\in T}\varphi (g^{-1}\gamma h)$. Since G is locally compact, the sum of

this series, denoted by $k_{\varphi}(g, h)$, is a continuous function on $G\times G$.

Let $g\in G$ and let $(\gamma , \eta )\in \Gamma \times \Gamma$. One has $k_{\varphi}(\gamma g, \eta h) =k_{\varphi}(g, h)$, hence $k_{\varphi}$ defines by passing to the quotient a continuous function on $X\times X$, which is denoted by $\widetilde{k}_{\varphi}$. One has moreover $\widetilde{k}_{\varphi}\in \mathscr{L}^2(X\times X,\widetilde{\mu}\otimes \widetilde{\mu})$, since the space X is assumed compact.

One denotes by $\dot{x}$ the image in X of an element $x$ of G under the canonical projection. Let $\varphi \in \mathscr{K}(G)$. For $f\in \mathscr{L}^2(X,\widetilde{\mu})$ and $x\in G$, one has

(INT, VII, p. 46, § 2, No.$^o3$, Prop. 5). Since $\widetilde{k}_{\varphi}$ belongs to the space $\mathscr{L}^2(X\times X,\widetilde{\mu}\otimes \widetilde{\mu})$, it follows that the endomorphism $\varrho (\varphi )$ of $L^2(X,\widetilde{\mu})$ coincides with the Hilbert-Schmidt endomorphism of kernel $\widetilde{k}_{\varphi}$; this endomorphism is therefore compact (Cor. 1 of III, p. 33)

There exists a sequence $(\varphi_n)_{n\in\mathbf{N}}$ of functions in $\mathscr{K}_+(G)$ of integral 1 such that $\varphi_n\cdot \mu$ converges to $\varepsilon_e$ in $\mathscr{M}_c(G)$ endowed with the topology of compact convergence (INT, VIII, p. 139, § 2, No.$^o7$, Cor. 2). Prop. 2 therefore implies that the representation $\varrho$ is semisimple and that the multiplicities of the irreducible unitary representations of G in $\varrho$ are finite.

The irreducible unitary representations of G whose multiplicity in $\varrho$ is not zero are called Γ-automorphic representations of the group G.