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

David C. Luo, Shaun Stevens

TL;DR

This work constructs and analyzes depth $\frac{1}{N}$ minimax (middle) supercuspidal representations of $\mathrm{GL}(2N, F)$ via Bushnell–Kutzko type theory. It shows that middle supercuspidals can be uniquely identified by their twisted gamma factors when twisted by tamely ramified quasi-characters of $F^{\times}$ and by simple supercuspials of $\mathrm{GL}(N,F)$, provided central characters are matched; a finite twist set $\Xi_{\text{middle}}$ suffices to control central data. The authors give explicit Whittaker and Bessel-type constructions for these representations, derive how the gamma factors depend on the triple $(\bar{f},\chi,\zeta)$, and demonstrate a refined local converse phenomenon at depth $\frac{1}{N}$ that improves upon the general bound. They also describe the Langlands parameter in tamely ramified cases (when $p mid 2N$) as an induced representation $\operatorname{Ind}_{E_f/F}\xi_{(f,\chi,\zeta)}$, and discuss a potential broader conjecture for refining local converse in this setting. The results offer a framework for distinguishing middle from non-middle depth representations and suggest a path to extending local converse theorems to broader families of supercuspidals.

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 $\text{GL}(N, F)$.

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

TL;DR

This work constructs and analyzes depth minimax (middle) supercuspidal representations of via Bushnell–Kutzko type theory. It shows that middle supercuspidals can be uniquely identified by their twisted gamma factors when twisted by tamely ramified quasi-characters of and by simple supercuspials of , provided central characters are matched; a finite twist set suffices to control central data. The authors give explicit Whittaker and Bessel-type constructions for these representations, derive how the gamma factors depend on the triple , and demonstrate a refined local converse phenomenon at depth that improves upon the general bound. They also describe the Langlands parameter in tamely ramified cases (when ) as an induced representation , and discuss a potential broader conjecture for refining local converse in this setting. The results offer a framework for distinguishing middle from non-middle depth representations and suggest a path to extending local converse theorems to broader families of supercuspidals.

Abstract

In this paper, we use type theory to construct a family of depth minimax supercuspidal representations of 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 and simple supercuspidal representations of .

Paper Structure

This paper contains 16 sections, 23 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and background
  3. Notation
  4. Maximal simple types
  5. Explicit Whittaker functions
  6. Simple supercuspidal representations
  7. Twisted local factors
  8. Central characters
  9. Middle supercuspidal representations
  10. Constructing middle supercuspidal representations
  11. Explicit Whittaker functions for middle supercuspidal representations
  12. Twisting by quasi-characters of $F^{\times}$
  13. Twisting by simple supercuspidal representations of $\mathop{\mathrm{GL}}\nolimits(N, F)$
  14. Alternate supercuspidal representations of depth $\frac{1}{N}$
  15. Conductor formulas
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\pi_{1}$ and $\pi_{2}$ be irreducible supercuspidal representations of $\mathop{\mathrm{GL}}\nolimits(n, F)$. If as functions of the complex variable $s$, for all irreducible supercuspidal representations $\tau$ of $\mathop{\mathrm{GL}}\nolimits(r, F)$ with $1 \leq r \leq \left\lfloor \frac{n}{2} \right\rfloor$, then $\pi_{1} \cong \pi_{2}$.

Theorems & Definitions (45)

  • Theorem : Local converse theorem
  • Theorem 1.1
  • Proposition 1.2
  • Conjecture 1.3
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 35 more