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

Abstract

Let $E/F$ be a quadratic extension of $p$-adic fields and ${\text{U}}_{2r+1}$ be the unitary group associated with $E/F$. We prove the following local converse theorem for ${\text{U}}_{2r+1}$: given two irreducible generic supercuspidal representations $\pi ,{\pi }_0$ of ${\text{U}}_{2r+1}$ 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 ${\text{GL}}_n(E)$ 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.

Accepted for publication in Transaction of the AMS