  Let $\Gamma$ be a Gromov hyperbolic group, endowed with an arbitrary
left-invariant hyperbolic metric, quasi-isometric to a word metric. The action
of $\Gamma$ on its boundary $\partial\Gamma$ endowed with the
Patterson-Sullivan measure $\mu$, after an appropriate normalization, gives
rise to a faithful unitary representation of $\Gamma$ on
$L^2(\partial\Gamma,\mu)$. We show that these representations are irreducible,
and give criteria for their unitary equivalence in terms of the metrics on
$\Gamma$. Special cases include quasi-regular representations on the Poisson
boundary.
