On Borel’s Regularization Theorem

Abstract

Introduction. Let G be a non compact semi simpel real Lie group with finite center and finitely many connected components. Assume that Γ ⊂ G is an arithmetic subgroup such that Γ\G is non compact. If M is a finite dimensional representation of G, then Borel proves that there are natural isomorphisms H(Γ,M) −→ Hj(Ω∗(X, M)umg) for all j, see [B 1, B 2, B 3, B 4]. Here Hj(Γ,M) is the group cohomology of Γ acting on M and Ω∗(X, M)umg is the complex of smooth M–valued Γ– invariant differential forms of uniform moderate growth on the symmetric space X attached to G. For the definition of growth conditions, see 1.1 and 2.4. The main result of this paper, theorem 3.2, says that the above isomorphy holds if Γ is any discrete subgroup of a reductive Lie group G with finitely many connected components. Moreover M can be any smooth representation of G on a Frechet space. Borel uses in his proof the Borel–Serre compactification of Γ\X and a tricky spectral–sequence argument to go from moderate growth conditions to uniform moderate growth conditions. In this paper we work essentially only on the symmetric space X. Here the uniform moderate growth conditions for the inverse map log of the exponential map exp p ∼ −→ X are most important, see 2.2 and 2.3. They are used to prove a global version of Poincare’s lemma with growth conditions on X, see 2.4. There are other contributions to Borel’s regularization theorem. J. Franke gives a different proof of Borel’s result in the adelic context for reductive algebraic groups, see [F]. U. Bunke and M. Olbrich give an extension of the first version of Borel’s result with moderate instead of uniform moderate growth conditions on Ω∗(X, M) to infinite dimensional coefficients M of moderate growth. Their result also holds for certain non arithmetic subgroups Γ of G, see [B–O].