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

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.