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

Abstract

In this paper, we prove the local converse theorem for $${\mathrm {Sp}}_{2r}(F)$$ over a p-adic field F. More precisely, given two irreducible supercuspidal representations of $${\mathrm {Sp}}_{2r}(F)$$ 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 $${\mathrm {GL}}_n(F)$$ 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 $${\mathrm {U}}(r,r)$$ .