  We consider Abelian covers of compact hyperbolic surfaces. We establish an
asymptotic expansion of the correlations for the horocycle flow on
$\mathbb{Z}^d$-covers, thus proving a strong form of Krickeberg mixing. We also
prove that the spectral measures around $0$ of the Casimir operators on any
increasing sequence of finite Abelian covers converge weakly to an absolutely
continuous measure.
