Automorphic Forms Ical Atom

We begin by quick recollection of the representation theory of GL(2) over a p-adic field, then introduce the double cover of GL(2) and provide both general results concerning the representations of this group and discuss principal series representation and the Weil representation, which yields in particular
a cuspidal representation under some conditions. We will mainly follow the following texts: (1) Gelbart: Weils representation and the spectrum of the metaplectic group, LNM 530 (2) Gelbart and Piatetski-Shapiro: Distinguished Representations and Modular forms of half-integral weight, Invent. Math, 59, 1990, 145-188.

This is a second talk in a series on representations of double cover of the p-adic points of GL(2) And SL(2). We will discuss irreducibility of principal series and existence of Whittaker Models for genuine representations of double cover of GL(2). If time permits, Eitan will start his lectures on local Langlands program for covering groups.

Following McNamara we describe n-fold covers of a split reductive group. We discuss splitting properties of these covers when restricted to unipotent and maximal compact subgroups. Since inverse image of a maximal torus is a Heisenberg group, we parametrise its (genuine) representations by their (genuine) central characters. Then we define principal series representation and study spherical vectors in these representation. Reference:- Peter McNamara; Principal Series Representations of Metaplectic Groups Over Local Fields. Multiple Dirichlet Series, L-functions and Automorphic Forms, Birkhauser Progress in Math. 300 (2012) 299–327.

One of the pillars of the Langlands program is L-functions. We will recall Artin’s L-functions, then briefly describe the modern automorphic point of view. One of the tools frequently used in the study of group representations and L-functions is called a model. Roughly speaking, a model is a unique realization of a representation in a convenient space of functions on the group. We will present a novel model: the metaplectic Shalika model. This is the analog of the Shalika model of GL(2n) of Jacquet and Shalika. One interesting representation having this model is the so-called exceptional representation of Kazhdan and Patterson of a cover of GL(n), which is the analog of the Weil representation. This representation is truly exceptional. We will describe it and its role in the study of the symmetric square L-function, and related problems.