Asymptotics of rational representations for algebraic groups

We study the asymptotic behaviour of the cohomology of subgroups of an algebraic group G with coefficients in the various irreducible rational representations of G and raise a conjecture about it. Namely, we expect that the dimensions of these cohomology groups approximate the ²-Betti numbers of with a controlled error term. We provide positive answers when G is a product of copies of SL₂. As an application, we obtain new proofs of J. Lott's and W. L\"uck's computation of the ²-Betti numbers of hyperbolic 3-manifolds and W. Fu's upper bound on the growth of cusp forms for non totally real fields, which is sharp in the imaginary quadratic case.