On the Analytic Continuation of c-Functions

Abstract

Let G be a reductive Lie group; Γ a nonuniform lattice in G. Then the theory of Eisenstein series plays a major role in the spectral decomposition of L2(G/Γ) (cf. [5]). One of the most difficult aspects of the subject is the analytic continuation of the Eisenstein series along with its associated c-function. This was originally done by Langlands using some very difficult analysis (cf. [5]). Later Harish-Chandra was able to simplify somewhat the most difficult part of the continuation, the continuation to zero, by the introduction of the Maas-Selberg relation. The purpose of this note is to give a simplified account of this particular part of the theory.Our chief tool will be the truncation operator of Arthur (cf. [1] and [8]), the systematic utilization of which has the effect of streamlining the earlier accounts, especially in so far as continuation to zero is concerned, which is reduced to an elementary manipulation.