On the dimension of the Jacquet module of a certain induced representation (Q2741187)

The purpose of this paper is to determine a certain Jacquet module of a certain induced representation of the \(n\)-fold metaplectic cover of \(\text{GL}(n,F)\), where \(F\) is a local field containing \(n\) \(n\)-th roots of 1. The representation is induced from a non-cuspidal exceptional representation on a parabolic subgroup of type \((n-1,1)\) and the Jacquet module is with respect to the minimal parabolic subgroup. This Jacquet module is a representation of the \(n\)-fold cover of the split Cartan subgroup. What the authors essentially prove here is that this representation consists of \(n\) irreducible components, which they determine explicitly. The proof consists of an application of the methods of \textit{I. N. Bernstein} and \textit{A. V. Zelevinsky} [Ann. Sci. Éc. Norm. Supér. (4) 10, 441-472 (1977; Zbl 0412.22015)] as developed for this case by \textit{D. A. Kazhdan} and \textit{S. J. Patterson} [Publ. Math., Inst. Hautes Étud. Sci. 62, 419 (1985; Zbl 0578.10034)].NEWLINENEWLINEFor the entire collection see [Zbl 0964.00058].