# Representation theory and the cohomology of arithmetic groups

### Birgit Speh

Cornell University, Ithaca, United States

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## Abstract

Let *G* be a semisimple Lie group with finitely many connected components and Lie algebra **g**, *K* a maximal compact subgroup of *G*, and *X* = *G*/*K* a symmetric space. A torsion free discrete subgroup Γ of *G* and a finite dimensional real or complex linear representation (*ρ*, *E*) of *G* define a locally symmetric space _X_Γ = Γ\*G*/*K* with a local system *Ẽ*. Then _H_∗ (Γ, *E*) = _H_∗ (Γ\*X*, *Ẽ*) is isomorphic to _H_∗ (**g**,*K*, _C_∞( Γ\*G*) ⊗ *E*). If Γ is an arithmetic group, then _H_∗ (**g**,*K*, _C_∞(Γ\*G*) ⊗ *E*) is isomorphic to the (**g**,*K*)-cohomology with coefficients in **A**(Γ\*G*) ⊗ *E* where **A**(Γ\*G*) is the space of automorphic forms. Using representation theory and the theory of automorphic forms a large amount of information about _H_∗ (Γ, *E*) can be deduced.