A conjecture about the analytical behaviour of Eisenstein series (Q2471450)

Let \(\Gamma\) be an arithmetic subgroup of a reductive group \(G\). \textit{R. P. Langlands} [On the functional equations satisfied by Eisenstein series. Lecture Notes in Mathematics. 544. Berlin etc.: Springer-Verlag (1976; Zbl 0332.10018)] has given an explicit spectral decomposition of the space \(L^2(\Gamma\backslash G)\) with respect to the right translation action of \(G\). The space decomposes into the space of cusp forms on which the spectrum is discrete and the orthogonal complement, which is spanned by Eisenstein series and contains discrete and continuous spectrum. In his argument, Langlands had to use contour movements which became quite intricate by the fact that the analytical behaviour of Eisenstein series is not completely understood. In the present paper, the author states a conjecture which would remedy these problems, yield a new proof of Langlands's theorem and allow stronger results like Paley-Wiener Theorems for Schwartz functions on \(\Gamma\backslash G\). He explains, how this conjecture can be deduced from known results in the case of cusp forms.