Residual automorphic forms and spherical unitary representations of exceptional groups (Q1955953)

Arthur's description of the discrete automorphic spectrum of an adèlic classical group via endoscopy, obtained using the trace formula, is now almost complete [\textit{J. Arthur}, The endoscopic classification of representations: orthogonal and symplectic groups. Providence, RI: American Mathematical Society, to appear (2013); \textit{C. P. Mok}, ``Endoscopic classification of representations of quasi-split unitary groups'', \url{arXiv:1206.0882}, Mem. Am. Math. Soc., to appear], although it is still conjectural subject to the stabilization of the twisted trace formula for the general linear group. As a part of his conjectures, \textit{J. Arthur} [Astérisque 171--172, 13--71 (1989; Zbl 0728.22014)] has suggested that unramified constituents of local principal series representations at a certain point of reducibility are unitary. This conjecture also makes sense for exceptional groups. The idea behind this claim is a simple observation that a local component of a discrete spectrum automorphic representation must be unitary, as it appears discretely in the \(L^2\)-spectrum. This observation was first used by \textit{B. Speh} [Math. Ann. 258, 113--133 (1981; Zbl 0483.22005)] to prove the unitarizability of certain representations of general linear groups over the reals, and later by \textit{M. Tadić} in the appendix of [Ann. Sci. Éc. Norm. Supér. (4) 19, No. 3, 335--382 (1986; Zbl 0614.22005)] for general linear groups over \(p\)-adic fields. There is another way to construct automorphic representations. It uses the Langlands spectral theory [\textit{R. P. Langlands}, On the functional equations satisfied by Eisenstein series. Berlin-Heidelberg-New York: Springer-Verlag (1976; Zbl 0332.10018)], which implies that certain residues of Eisenstein series span automorphic representations in the residual spectrum. Hence, Arthur's conjecture on the unitarizability of unramified constituents of principal series representation can be proved by constructing an Eisenstein series, which is everywhere unramified, and shows that it has a pole with a square-integrable residue at the point suggested by Arthur. This method has been already applied for the general linear groups [\textit{H. Jacquet}, Lect. Notes Math. 1041, 185--208 (1984; Zbl 0539.22016); \textit{C. Moeglin} and \textit{J.-L. Waldspurger}, Ann. Sci. Éc. Norm. Supér. (4) 22, No. 4, 605--674 (1989; Zbl 0696.10023)], other classical groups [\textit{C. Moeglin}, Forum Math. 6, No. 6, 651--744 (1994; Zbl 0816.11034)], and the exceptional Chevalley group \(G_2\) [\textit{H. H. Kim}, Can. J. Math. 48, No. 6, 1245--1272 (1996; Zbl 0879.11024); \textit{S. Žampera}, J. Math. Pures Appl., IX. Sér. 76, No. 9, 805--835 (1997; Zbl 0886.11030)]. The paper under review applies the same method to the remaining cases, that is, the Chevalley groups of type \(E_6\), \(E_7\), \(E_8\), and \(F_4\), and thus proves that the unramified constituents in question are indeed unitarizable. The crucial obstacle in the study of the analytic properties of the everywhere unramified Eisenstein series on these exceptional groups is the complicated combinatorics and possible cancellations in the expression for the constant term of the Eisenstein series as a sum over the Weyl group (which is large) of certain ratios of the Dedekind zeta functions of the underlying number field. The author overcomes this difficulty by replacing the point suggested by Arthur with another point of a special form in the same Weyl group orbit. This does not change the unramified constituent, and the form of this new point of evaluation for the Eisenstein series simplifies the computation considerably. Essentially it reduces the computation to the case of the Eisenstein series associated to the trivial representation of a maximal proper parabolic subgroup. Even after this reduction, the explicit calculation has to be done by a computer. The author used exact symbolic calculation in \texttt{Mathematica 8}, and the programs are available for download as an on-line supplement to the paper. The computer calculations required further rewriting of the problem using some software tricks.