Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A local converse theorem for quasi-split even special orthogonal groups

Alexander Hazeltine

TL;DR

The paper provides a direct local converse theorem for the quasi-split non-split even special orthogonal group SO_{2l} over a non-Archimedean field of characteristic not 2, employing Howe vectors and partial Bessel functions to compare twisted γ-factors γ(s, π × τ, ψ) with twists by GL_n (n ≤ l). By developing a detailed framework for partial Bessel functions in the quasi-split setting and analyzing their behavior under the outer automorphism c, the authors show that equal twisted γ-factors determine the representation π up to outer conjugacy (π ≅ π' or π ≅ π'^c). The approach provides an intrinsic, representation-theoretic route that parallels results for split cases and is robust for both supercuspidal and generic representations, including implications for automorphic rigidity. The work advances the understanding of how twisted γ-factors encode information about representations on SO_{2l}, highlighting the role of outer automorphisms and the finer structure of partial Bessel functions in non-split quasi-split groups.

Abstract

We give a direct proof of the local converse theorem for quasi-split non-split $\mathrm{SO}_{2l}$ over a local non-Archimedean field of characteristic $p\neq 2$, applying the theory of Howe vectors and partial Bessel functions.

A local converse theorem for quasi-split even special orthogonal groups

TL;DR

The paper provides a direct local converse theorem for the quasi-split non-split even special orthogonal group SO_{2l} over a non-Archimedean field of characteristic not 2, employing Howe vectors and partial Bessel functions to compare twisted γ-factors γ(s, π × τ, ψ) with twists by GL_n (n ≤ l). By developing a detailed framework for partial Bessel functions in the quasi-split setting and analyzing their behavior under the outer automorphism c, the authors show that equal twisted γ-factors determine the representation π up to outer conjugacy (π ≅ π' or π ≅ π'^c). The approach provides an intrinsic, representation-theoretic route that parallels results for split cases and is robust for both supercuspidal and generic representations, including implications for automorphic rigidity. The work advances the understanding of how twisted γ-factors encode information about representations on SO_{2l}, highlighting the role of outer automorphisms and the finer structure of partial Bessel functions in non-split quasi-split groups.

Abstract

We give a direct proof of the local converse theorem for quasi-split non-split over a local non-Archimedean field of characteristic , applying the theory of Howe vectors and partial Bessel functions.

Paper Structure

This paper contains 11 sections, 31 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. The groups and representations
  3. Zeta integrals and the gamma factor
  4. Partial Bessel functions
  5. Howe vectors
  6. Conjugate of partial Bessel Function
  7. Bessel Support
  8. Bruhat Order
  9. Outline of the proof of Theorem \ref{['converse thm intro']}
  10. Sections of induced representations
  11. Proof of the local converse theorem

Key Result

Theorem 1.2

Let $F$ be a non-Archimedean field of characteristic $p\neq 2$ and $\pi$ and $\pi^\prime$ be irreducible $\psi$-generic supercuspidal representations of quasi-split non-split $\mathrm{SO}_{2l}(F)$ with the same central character $\omega$. If for all irreducible generic representations $\tau$ of $\mathrm{GL}_n(F)$ with $n\leq l,$ then $\pi\cong\pi'$ or $\pi\cong \pi'^c,$ where $c$ is the outer aut

Theorems & Definitions (47)

  • Conjecture 1.1: Local Converse Theorem for $\mathrm{GL}_l$
  • Theorem 1.2: The Local Converse Theorem for quasi-split non-split $\mathrm{SO}_{2l}$, supercuspidal case
  • Theorem 1.3: The Local Converse Theorem for quasi-split non-split $\mathrm{SO}_{2l}$, generic case
  • Theorem 1.4: Theorem \ref{['l thm']}
  • Theorem 1.5
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 37 more