On the Ramanujan conjecture for quasisplit groups (Q2484143)

The paper surveys recent results on the Ramanujan conjecture until the year 2004. The most general version of Ramanujans conjecture, which the author states, is that the local component \(\pi_v\) (at a place \(v\) of a global field \(F\)) of every globally generic cuspidal representation \(\Pi\) of the \(F\)-adelic points of a quasisplit connected reductive algebraic group \(G\) must be a tempered representation of the group of \(F_v\)-rational points of \(G\). Considering the Langlands classification for an irreducible admissible local component \(\pi\), by \textit{A. J. Silberger} [Math. Ann. 236, 95--104 (1978; Zbl 0362.20029)], Theorem 4.1 (3) in the \(p\)-adic case, or \textit{A. Borel} and \textit{N. Wallach} [Continuous cohomology, discrete subgroups, and representations of reductive groups (Ann. of Math. Stud. 94, Princeton Univ. Press 1980; Zbl 0443.22010)], Theorem 4.11 in the Archimedean case, \(\pi\) is given by a Langlands quotient parametrized by a tempered representation \(\sigma\) of a suitable Levi subgroup of \(G\) and a parameter \(\nu\) which determines a suitable character twist within a normalized parabolic induction. The conjecture \(\pi= \sigma\) and \(\nu= 0\) is now reformulated by the assertion that \(\nu\) can be arbitrarily small bounded. Indeed for \(\text{GL}_n\) the parameter \(\nu\) can be formulated as a sequence of positive real numbers parametrizing unramified twists of \(\sigma\) (see \textit{S. S. Kudla} [Proc. Symp. Pure Math. 55, 365--391 (1994; Zbl 0811.11072)], Theorem 2.2.2). The author surveys the discussion on several bounds and additional reformulations for cases other than \(\text{GL}_n\). A large part of the paper surveys results concerning the Langlands functoriality because this method, for instance, makes it possible to reduce the conjecture for classical groups to that for \(\text{GL}_n\).