Classification of distinguished representations of GL(2)

TL;DR

This paper classifies irreducible representations of GL(2) over a non-archimedean local field that are distinguished by a maximal torus, using Casselman's criteria for p-adic symmetric spaces.

Contribution

It provides a classification of H-distinguished irreducible representations of GL(2) over non-archimedean fields, applying Casselman's criteria in this context.

Findings

Complete classification of H-distinguished representations

Application of Casselman's criteria to p-adic symmetric spaces

Advancement in understanding distinguished representations of GL(2)

Abstract

Let $F$ be a non-archimedean local field with odd residual characteristic, and let $H$ be a maximal torus of ${\rm GL}_2(F)$. In this paper, we will classify the irreducible $H$-distinguished representations of ${\rm GL}_2(F)$ by using Casselman's criteria for $p$-adic symmetric spaces developed by Kato-Takano.