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).