  Given a locally compact group $G$ and a unitary representation $\rho:G\to
U({\mathcal H})$ on a Hilbert space ${\mathcal H}$, we construct a
$C^*$-correspondence ${\mathcal E}(\rho)={\mathcal H}\otimes_{\mathbb C}
C^*(G)$ over $C^*(G)$ and study the Cuntz-Pimsner algebra ${\mathcal
O}_{{\mathcal E}(\rho)}$. We prove that for $G$ compact, ${\mathcal
O}_{{\mathcal E}(\rho)}$ is strong Morita equivalent to a graph $C^*$-algebra.
If $\lambda$ is the left regular representation of an infinite, discrete and
amenable group $G$, we show that ${\mathcal O}_{{\mathcal E}(\lambda)}$ is
simple and purely infinite, with the same $K$-theory as $C^*(G)$. If $G$ is
compact abelian, any representation decomposes into characters and determines a
skew product graph. We illustrate with several examples and we compare
${\mathcal E}(\rho)$ with the crossed product $C^*$-correspondence.
