Locally Symmetric Spaces Seminar

Let $H = G/K$ be a symmetric space and $X = \Gamma \backslash H$ its locally symmetric quotient. An important problem is to understand the cohomology of the space $X$ (or, more or less equivalent, cohomology of the group $\Gamma$). The idea is that on this system of cohomology groups for various subgroups $\Gamma$ we have action of Hecke algebras that give us an automorphic structure on this system.As we have seen, in case when the space $X$ is compact, the cohomology groups $H^*(X, \mathbf C)$ can be computed in terms of automorphic representations. In non-compact case the computations become very involved.In my informal talk I will try to suggest that in this approach we have to replace the system of cohomology groups $H^*(X)$ by closely related but different system of groups $H_t^*(X)$. This system of groups also has appropriate Hecke properties, but it is much easier to relate these groups to automorphic representations.One of the standard attempts to do this is to use $L^2$ cohomology. I think this is not correct approach. I will propose a notion of tempered cohomology, that I believe is the correct replacement for $L^2$ cohomology. I conjecture that these cohomology groups have nice properties we expect from them.