Representation theory and the cohomology of arithmetic groups

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.