A Remark on Eisenstein Series

The theory of Eisenstein series is fundamental for the spectral theory of automorphic forms. It was first developed by Selberg, and was completed by Langlands ([Lan76]; see also [MW95]). There are several known proofs for the meromorphic continuation of Eisenstein series (apart from very special cases of Eisenstein series which can be expressed in terms of Tate integrals). In all these proofs it is convenient, if not essential, to assume (in the number field case) that the inducing section is K-finite, to ensure finite dimensionality. However, the analytic properties of Eisenstein series are closely tied to, and at any rate controlled by, those of the intertwining operators. The latter decompose into local intertwining operators. In the archimedean case, a lot is known about the local intertwining operators and no K-finiteness assumption on the section is necessary. It is therefore reasonable to expect that the analytic properties of Eisenstein series for a general smooth section follow from that of K-finite sections. The modest goal of this short note is to carry this out (using the meromorphic continuation of K-finite Eisenstein series as a black box). In fact, by the automatic continuity theorem of Casselman and Wallach ([Wal92, Ch. 11]), at each regular point the Eisenstein series can be extended to smooth sections. This by itself does not suffice to prove meromorphic continuation unless one knows some local uniformity (in the spectral parameter) for the modulus of continuity of the Eisenstein series as a map from the induced representation to the space of automorphic forms. The point is that such uniformity, at least for cuspidal Eisenstein series, is provided by the Maass-Selberg relations together with the properties of the intertwining operators at the archimedean places. As for Eisenstein series induced from other discrete spectrum, their properties can be deduced from those of cuspidal Eisenstein series by Langlands’ general theory. We remark however, that there is still an