On the Local Converse Theorem for Depth $\frac{1}{N}$ Supercuspidal Representations of $\text{GL}(2N, F)$

TL;DR

This paper constructs a new family of depth 1/N supercuspidal representations of GL(2N, F) using type theory, and demonstrates their uniqueness via explicit gamma factor computations, extending simple supercuspidals.

Contribution

It introduces middle supercuspidal representations as a natural generalization of simple supercuspidals and establishes their uniqueness through explicit gamma factor analysis.

Findings

Constructed middle supercuspidal representations of depth 1/N.

Showed these representations are uniquely determined by twisting with quasi-characters.

Extended the understanding of supercuspidal representations beyond minimal positive depth.

Abstract

In this paper, we use type theory to construct a family of depth $\frac{1}{N}$ minimax supercuspidal representations of $\text{GL}(2N, F)$ which we call middle supercuspidal representations. These supercuspidals may be viewed as a natural generalization of simple supercuspidal representations, i.e. those supercuspidals of minimal positive depth. Via explicit computations of twisted gamma factors, we show that middle supercuspidal representations may be uniquely determined through twisting by quasi-characters of $F^{\times}$ and simple supercuspidal representations of $\text{GL}(N, F)$.