Analysis on Symmetric and Locally Symmetric Spaces (Multiplicities, Cohomology and Zeta functions)

The goal of the course was a review of results mainly due to M. Olbrich and the first author. We consider a discrete cocompact subgroup $\mathrm{\Gamma }$ of a semisimple Lie group $G$. We relate the group cohomology of $\mathrm{\Gamma }$ with coefficients in the maximal globalization of a representation of $G$ with the multiplicities of unitary representations of $G$ in $L^2(G\mathrm{/}\mathrm{\Gamma })$. Explicit calculations are given in the case $G=SL(2,R)$. In the rank-one case we state a version of the Selberg trace formula, discuss the singularities of the Selberg zeta-functions, and state their relation with the cohomology of $\mathrm{\Gamma }$ in principal series representations.