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

TL;DR

This paper proves a local converse theorem for odd special orthogonal and symplectic groups over positive characteristic fields, establishing conditions under which two generic representations are isomorphic based on gamma factors.

Contribution

It extends the local converse theorem to positive characteristic fields for these groups, using partial Bessel functions and classification of generic representations.

Findings

Established the local converse theorem for SO and Sp groups in positive characteristic.

Extended the rank of twists for gamma factors up to 2r-1.

Generalized previous results from characteristic zero to positive characteristic.

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 $0$.