Analytic continuation and functional equations of cuspidal Eisenstein series for maximal cuspidal subgroups

In this work we develop an approach of Selberg to the analytic continuation of Eisenstein series by means of Fredholm theory. We set up the machinery for very general algebraic groups and arithmetic subgroups, and obtain the analytic continuation and functional equations of cuspidal Eisenstein series for rank one cuspidal subgroups. This approach suggests an alternative development to part of Langlands' theory of Eisenstein series. It also opens up the possibility of more precise knowledge about the poles of Eisenstein series.