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

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

2005-08-29

arxiv.org/abs/math/0508560v1

Mathematics

Representation Theory

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

Scientific paper

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.

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