Classification of standard modules with linear periods

TL;DR

This paper classifies standard modules of GL_n(D) with non-zero linear periods over non-Archimedean fields, simplifying the Prasad-Takloo-Bighash conjecture to essentially square-integrable cases.

Contribution

It provides a classification of standard modules with linear periods, reducing the conjecture to a more manageable subclass of representations.

Findings

Classification of standard modules with linear periods

Reduction of the Prasad-Takloo-Bighash conjecture

Focus on essentially square-integrable representations

Abstract

Suppose that $F$ is a non-Archimedean local field and $D$ is a central division algebra over $F$. Let $n$ be a positive integer. We show a classification modulo essentially square-integrable representations of standard modules of $\mathrm{GL}_n(D)$ which have non-zero linear periods. By this classification, the conjecture of Prasad and Takloo-Bighash is reduced to the case of essentially square integrable representations.