# A local converse theorem for $\textrm{U}_{2r+1}$

## TL;DR

This paper establishes a local converse theorem for odd-dimensional unitary groups over p-adic fields, showing that gamma factors determine irreducible generic supercuspidal representations.

## Contribution

It proves a new local converse theorem for $ extrm{U}_{2r+1}$, linking gamma factors with representation equivalence, using analysis of local integrals and partial Bessel functions.

## Key findings

- Gamma factors determine irreducible supercuspidal representations.
- The proof utilizes properties of partial Bessel functions.
- The theorem applies to all $n$ with $1 \\le n \\le r$.

## Abstract

Let $E/F$ be a quadratic extension of $p$-adic fields and $\textrm{U}_{2r+1}$ be the unitary group associated with $E/F$. We prove the following local converse theorem for $\textrm{U}_{2r+1}$: given two irreducible generic supercuspidal representations $\pi,\pi_0$ of $\textrm{U}_{2r+1}$ with the same central character, if $\gamma(s,\pi\times \tau,\psi)=\gamma(s,\pi_0\times \tau,\psi)$ for all irreducible generic representation $\tau$ of $\textrm{GL}_n(E)$ and for all $n$ with $1\le n\le r$, then $\pi\cong \pi_0$. The proof depends on analysis of the local integrals which define local gamma factors and uses certain properties of partial Bessel functions developed by Cogdell-Shahidi-Tsai recently.