On Speh representations for level zero supercuspidal representations and Ginzburg-Kaplan gamma factors

TL;DR

This paper explores the relationship between Speh representations over finite fields and local fields, linking their Whittaker models and gamma factors through level zero supercuspidal representations.

Contribution

It establishes a connection between Speh representations over finite fields and local fields, and relates their gamma factors via a commutative diagram involving Whittaker models.

Findings

Relation between Speh representations over finite fields and local fields.

Connection of Whittaker models for these representations.

Linking Ginzburg-Kaplan gamma factors across settings.

Abstract

We establish a relation between Speh representations of $\mathrm{GL}_n\left(\mathbb{F}_q\right)$ and Speh representations of $\mathrm{GL}_n\left(F\right)$, where $F$ is a non-archimedean local field. We use irreducible level zero supercuspidal representations to show that these two notions of Speh representations associated to cuspidal representations are related via a commutative diagram, and that their corresponding $(k,c)$ $\psi$-Whittaker models are also related. We use these results to relate the local Ginzburg-Kaplan integrals for level zero supercuspidal representations to their finite field counterparts.