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

Abstract

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 $\Gamma$ of a semisimple Lie group $G$. We relate the group cohomology of $\Gamma$ with coefficients in the maximal globalization of a representation of $G$ with the multiplicities of unitary representations of $G$ in $L^2(G/\Gamma )$. 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 $\Gamma$ in principal series representations.

11 pages, Notes of a course given at the Summer School "Algebraic Groups" in Goettingen, June 27 - July 15