Quaternionic Speh representations

TL;DR

This paper introduces quaternionic Speh representations for $ ext{GL}_{k,D}$, analyzing their models locally and their Fourier coefficients globally, advancing understanding of their structure and automorphic properties.

Contribution

It defines and studies quaternionic Speh representations, providing new insights into their models and Fourier coefficients compared to classical Speh representations.

Findings

Unique models for local representations established

Non-vanishing Fourier coefficients demonstrated globally

Analogues of Speh representations in quaternionic setting

Abstract

For a central division algebra $D$, we study a family of representations of $\mathrm{GL}_{k,D}$ (both locally and globally), which can be viewed as analogues of the Speh representations. Locally, we study unique models for these representations. Globally, we show that these representations support certain non-vanishing Fourier coefficients.