Let \(G\subset \mathrm{GL}(n,\mathbb{R})\) be a real reductive algebraic group subject to the assumptions in Harish-Chandra [Automorphic forms on semisimple Lie groups, Lect. Notes Math. 62. Berlin etc.: Springer Verlag (1968; Zbl 0186.04702), 103–106]; for some results additional assumptions on \(G\) are required. Let \(\Gamma\) be an arithmetical subgroup of \(G\).
The book under review presents new proofs of the meromorphic continuation and the functional equation of the Eisenstein series associated to cusp forms on the Levi-component of a rank one cuspidal parabolic subgroup of \(G\). As is well-known, these play an important role in the spectral decomposition of \(L^2(\Gamma \setminus G)\). The basic idea of the present work is due to A. Selberg. The proofs are based on a suitable version of the Selberg principle that the Eisenstein series under consideration satisfy a certain convolution equation (with compactly supported kernel function) and the reduction of the problem via truncation to the investigation of a Fredholm integral equation.
A very readable introduction of 38 pages contains the description of some background, a rough exposition of the main ideas of the proof as well as a discussion of the relations to other proofs and methods. The author notes that “the conceptual simplicity of the present approach invites one to attempt the development with cuspidal Eisenstein series for higher rank cuspidal subgroups”.
Moreover, the question arises if a further development of the present method leads to an alternative proof of the Langlands theory of the spectral decomposition of \(L^2(\Gamma \setminus G)\).