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In this paper we extend the construction of special representations toconvex-cocompact isometry groups of CAT(-1) spaces which admits complementaryseries. We prove that these limits of complementary series have a naturalnon-vanishing reduced cohomology class [c]. As a by product we generalize KuhnVershik formula and characterize geometrically CAT(-1) groups that admitcomplementary series. Investigating dynamical properties of the cohomologyclass [c] we prove an cocycle equidistribution theorem \`a la Roblin-Margulisand deduce the irreducibility of the associated affine actions. Theconstructions and results apply beyond the class of rank 1 linear groups wherethe irreducibility of the affine actions associated to the canonical class [c],even in the case of uniform lattices in SO(n,1), SU(n,1) or SL2(Qp) with n>1and p prime, had not been established.

Limits of complementary series, special representations and cohomology of CAT(-1) groups