A corollary to Bernstein's theorem and Whittaker functionals on the metaplectic group
This paper begins by reviewing a theorem of Bernstein that is frequently useful in addressing analytic continuation and rationality questions in the setting of automorphic forms and \(L\)-functions. Additionally, a corollary is proved that sometimes gives more precise information about the location of poles. This corollary is then used to check the analytic continuation of a certain set of Whittaker functionals on the metaplectic group. More precisely, let \({\mathbb F}\) be a nonarchimedean local field containing the \(n\)-th roots of unity and such that \(|n|_{\mathbb F}=1.\) Write \(G=GL_r({\mathbb F})\) and \(\widetilde{G}\) for the \(n\)-fold metaplectic cover of \(G\). Let \(\pi\) be an admissible representation of the cover of a standard Levi subgroup of type \((r_1, \ldots, r_k)\) having \(l>0\) linearly independent Whittaker functionals. For each \(s\in{\mathbb C}^k\), \(\pi\) may be twisted by an unramified quasicharacter and then induced to obtain a representation, \(\pi_s\), of \(\widetilde{G}\). The author has already shown that \(\pi_s\) has \(l\) linearly independent Whittaker functionals. The main result here shows that these functionals may be chosen to vary holomorphically with \(s\in{\mathbb C}^k\). This result is a first step for a general theory of local coefficients for the metaplectic group which, in turn, should contribute to the understanding of the conjectural Shimura correspondence from \(\widetilde{G}\) to \(G\).
scientific article; zbMATH DE number 1489472