A local converse theorem for $\textrm{Sp}_{2r}$

Abstract: In this paper, we prove the local converse theorem for $\textrm{Sp}{2r}(F)$ over a $p$-adic field $F$. More precisely, given two irreducible supercuspidal representations of $\textrm{Sp}{2r}(F)$ with the same central character such that they are generic with the same additive character and they have the same gamma factors when twisted with generic irreducible representations of $\textrm{GL}_n(F)$ for all $1\le n\le r$, then these two representations must be isomorphic. Our proof is based on the local analysis of the local integrals which define local gamma factors. A key ingredient of the proof is certain partial Bessel function property developed by Cogdell-Shahidi-Tsai recently. The same method can give the local converse theorem for $\textrm{U}(r,r)$.