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

TL;DR

This paper proves a local converse theorem for symplectic groups over p-adic fields, establishing conditions under which two supercuspidal representations are isomorphic based on gamma factors and local integrals.

Contribution

It introduces a new proof for the local converse theorem for $ extrm{Sp}_{2r}$ using local integrals and partial Bessel functions, extending methods to $ extrm{U}(r,r)$.

Findings

Proves the local converse theorem for $ extrm{Sp}_{2r}$ over p-adic fields.

Establishes isomorphism criteria based on gamma factors and local integrals.

Extends the method to the unitary group $ extrm{U}(r,r)$.

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 $\textrm{U}(r,r)$.