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The local converse theorem for odd special orthogonal and symplectic groups in positive characteristic

Yeongseong Jo

TL;DR

The paper proves a local converse theorem for irreducible generic representations of ${\rm SO}_{2r+1}(F)$ and ${\rm Sp}_{2r}(F)$ over non-archimedean local fields of positive characteristic (${\rm char}(F)\neq 2$). It develops and compares Rankin–Selberg gamma factors with Langlands–Shahidi gamma factors, then uses Cogdell–Shahidi–Tsai partial Bessel functions to deduce that equality of gamma factors twisted by all irreducible generic $\tau$ of ${\rm GL}_n(F)$ for $1\le n\le r$ forces representation isomorphism, extending prior results to all generic representations in positive characteristic. The work handles both the odd orthogonal and symplectic/metaplectic settings: for ${\rm SO}_{2r+1}(F)$ via Bessel models and for ${\rm Sp}_{2r}(F)$ via Fourier–Jacobi models and the Weil representation, establishing multiplicativity, stability, and a suitable local Langlands lift to ${\rm GL}_{2r}(F)$ or ${\rm GL}_{2r+1}(F)$. These results lay groundwork for local functorial transfers in positive characteristic and align with broader aims of transfer principles across classical groups.

Abstract

Let $F$ be a non-archimedean local field of characteristic different from $2$ and $G$ be either an odd special orthogonal group ${\rm SO}_{2r+1}(F)$ or a symplectic group ${\rm Sp}_{2r}(F)$. In this paper, we establish the local converse theorem for $G$. Namely, for given two irreducible admissible generic representations of $G$ with the same central character, if they have the same local gamma factors twisted by irreducible supercuspidal representations of ${\rm GL}_n(F)$ for all $1 \leq n \leq r$ with the same additive character, these representations are isomorphic. Using the theory of Cogdell, Shahidi, and Tsai on partial Bessel functions and the classification of irreducible generic representations, we break the barrier on the rank of twists $1 \leq n \leq 2r-1$ in the work of Jiang and Soudry, and extend the result of Q. Zhang, which was achieved for all supercuspidal representations in characteristic $0$.

The local converse theorem for odd special orthogonal and symplectic groups in positive characteristic

TL;DR

The paper proves a local converse theorem for irreducible generic representations of and over non-archimedean local fields of positive characteristic (). It develops and compares Rankin–Selberg gamma factors with Langlands–Shahidi gamma factors, then uses Cogdell–Shahidi–Tsai partial Bessel functions to deduce that equality of gamma factors twisted by all irreducible generic of for forces representation isomorphism, extending prior results to all generic representations in positive characteristic. The work handles both the odd orthogonal and symplectic/metaplectic settings: for via Bessel models and for via Fourier–Jacobi models and the Weil representation, establishing multiplicativity, stability, and a suitable local Langlands lift to or . These results lay groundwork for local functorial transfers in positive characteristic and align with broader aims of transfer principles across classical groups.

Abstract

Let be a non-archimedean local field of characteristic different from and be either an odd special orthogonal group or a symplectic group . In this paper, we establish the local converse theorem for . Namely, for given two irreducible admissible generic representations of with the same central character, if they have the same local gamma factors twisted by irreducible supercuspidal representations of for all with the same additive character, these representations are isomorphic. Using the theory of Cogdell, Shahidi, and Tsai on partial Bessel functions and the classification of irreducible generic representations, we break the barrier on the rank of twists in the work of Jiang and Soudry, and extend the result of Q. Zhang, which was achieved for all supercuspidal representations in characteristic .
Paper Structure (13 sections, 26 theorems, 113 equations)

This paper contains 13 sections, 26 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. The Local Converse Theorem for SO$(2r+1)$
  3. Structure theory of odd orthogonal groups
  4. Uniqueness of Bessel models
  5. The intertwining operator
  6. The local zeta integrals and gamma factors
  7. The equality of gamma factors
  8. The local converse theorem
  9. The Local Converse Theorem for Sp$(2r)$
  10. Structure theory of symplectic groups and Weil representations
  11. Uniqueness of Fourier--Jacobi models
  12. The local zeta integrals and gamma factors
  13. The local converse theorem

Key Result

Theorem A

Let $\pi$ and $\pi'$ be irreducible admissible generic representations of $G$ with the common central character $\omega$. If for all irreducible generic supercuspidal representations $\tau$ of ${\rm GL}_n(F)$ and for all $n$ with $1 \leq n \leq r$, then $\pi \cong \pi'$.

Theorems & Definitions (42)

  • Theorem A
  • Proposition 2.1: Uniqueness of Bessel models
  • proof
  • Proposition 2.2: Local functional equation
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • Theorem 2.5
  • proof
  • ...and 32 more