A New Proof to the Period Problems of GL(2)

TL;DR

This paper introduces a novel proof for the period problems of GL(2) over quadratic local fields, utilizing relations between base change, theta lifts, and Whittaker models, and classifies distinguished representations of D× over E.

Contribution

It provides a new proof for period problems of GL(2) and classifies distinguished representations of D× over quadratic extensions, advancing understanding of local and global representation theory.

Findings

New proof for period problems of GL(2)

Classification of D×(F)-distinguished representations

Relations between base change, theta lifts, and Whittaker models

Abstract

We use the relations between the base change representations, theta lifts and Whittaker model, to give a new proof to the period problems of $GL(2)$ over a quadratic local field extension $E/F.$ And we classify both local and global $D^\times(F)-$distinguished representations $\pi^D$ of $D^\times(E),$ where $D^\times$ is an inner form of $GL_2$ defined over a nonarchimedean field or a number field $F.$