Asymptotics of rational representations for algebraic groups

Abstract

We study the asymptotic behaviour of the cohomology of subgroups $\Gamma$ 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 L2-analogues of Betti numbers of $\Gamma$ with a controlled error term. We provide positive answers when $G$ is a product of copies of $S{L}_2$.

arXiv comment33 pages, comments are welcome!