Local Representations: The p-adic Case

Abstract

In this chapter we turn to the local non-archimedean case and study the representation theory of G J (F), where F is a finite extension of some \( {\mathbb{Q}_p} \).We will reach the goal of classifying all irreducible, admissible representations of G J (F) by using the fundamental relation \(\pi \simeq \tilde{\pi } \otimes \pi _{{SW}}^{m} \) and the classification of representations of the metaplectic group given by Waldspurger in [Wa1]. Roughly speaking, all non-supercuspidal irreducible representations of Mp are obtained by parabolic induction. This result can be taken over to the Jacobi group by making the isomorphism \(\pi \simeq \tilde \pi \otimes \pi _{SW}^m \) explicit in the context of standard models for induced representations (Theorem 5.4.2).KeywordsLocal RepresentationPrincipal SeriesCompact Open SubgroupAdmissible RepresentationPrincipal Series RepresentationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.