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

Abstract:

Let $E/F$ be a quadratic extension of $p$-adic fields, and let ${\mathrm {U}}_{2r+1}$ be the unitary group associated with $E/F$. We prove the following local converse theorem for ${\mathrm {U}}_{2r+1}$: given two irreducible generic supercuspidal representations $\pi ,\pi _0$ of ${\mathrm {U}}_{2r+1}$ with the same central character, if $\gamma (s,\pi \times \tau ,\psi )=\gamma (s,\pi _0\times \tau ,\psi )$ for all irreducible generic representations $\tau$ of ${\mathrm {GL}}_n(E)$ 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 defines local gamma factors and uses certain properties of partial Bessel functions developed recently by Cogdell, Shahidi, and Tsai.