Spectral decomposition of a covering of \(GL(r)\): The Borel case (Q2783393)

Let \(k\) be an algebraic number field containing the \(n\)th roots of unity. In this memoir the author analyzes the automorphic spectral theory of the \(n\)-fold metaplectic cover of \(GL_r(k_A)\). The methods used for achieving this end are introduced by \textit{R. P. Langlands} in [On the functional equations satisfied by Eisenstein series, Lect. Notes Math. 544 (1976; Zbl 0332.10018)] and developed further by \textit{C. Moeglin} and \textit{J.-L.Waldspurger} in [Ann. Sci. Éc. Norm. Supér. (4) 22, 605-674 (1989; Zbl 0696.10023) and Spectral decomposition and Eisenstein series, Prog. Math. 113 (1994; Zbl 0794.11022)]. Although the theory is analogous to the classical case, in the non-metaplectic case it is even more intricate and the author has made a start by treating the non-cuspidal case. Despite its complications, this work is basic for any attempt to prove the (conjectural) generalized Shimura correspondence by trace-theoretic methods.