A conjecture on Whittaker-Fourier coefficients of cusp forms
Let \(G\) be a quasi-split reductive group over a number field \(F\), \(B\) be a Borel subgroup defined over \(F\), \(N\) its unipotent radical, and \(\psi_N\) a non-degenerate character of \(N(F)\backslash N(\mathbb{A})\). If \(\pi\) is a \(\psi_N\)-generic cuspidal automorphic representation of \(G(\mathbb{A})\), then the authors introduce a quantity \(c_\pi^{\psi_N}\) which measures the size of the \(\psi_N\)-Whittaker-Fourier coefficient of a matrix coefficient of \(\pi\) divided by a suitable product of Whittaker coefficients for the corresponding forms in the spaces \(\pi\) and \(\pi^\vee\). The precise definition requires some regularizations, which are treated in the paper. Using Rankin-Selberg theory, the authors show that this quantity is \(1\) for irreducible cuspidal representations of \(GL_n\); a similar result was proved independently by \textit{Y. Sakellaridis} and \textit{A. Venkatesh} [Periods and harmonic analysis on spherical varieties. Paris: Société Mathématique de France (SMF) (2017; Zbl 1479.22016)]. They then provide a conjectured value for the quantity more generally, based on Arthur's conjectures for the discrete spectrum, in the case of elliptic Arthur parameters of Ramanujan type. They reduce their conjecture, on some natural compatibility assumptions on Arthur's conjecture, to the case that \(G\) is semisimple and simply connected. They also formulate the conjecture more concretely when \(G\) is quasi-split and classical, by tying it to descent, and formulate a version for the metaplectic double cover of the symplectic group. The conjectures are also proved in some low rank cases. The conjectures of this paper may be viewed as an analogue for Whittaker coefficients of the conjectures of \textit{A. Ichino} and \textit{T. Ikeda} [Geom. Funct. Anal. 19, No. 5, 1378--1425 (2010; Zbl 1216.11057)], which concern periods of automorphic forms over closed subgroups and which sharpen the global Gross-Prasad conjecture. They also strengthen the conjectures of Sakellaridis and Venkatesh for periods over spherical subgroups in this specific case.
scientific article; zbMATH DE number 6359743
scientific article