The local converse theorem for ${\rm SO}(2n+1)$ and applications

Abstract

In this paper we characterize irreducible generic representations of ${\rm SO}_{2n+1}(k)$ (where $k$ is a $p$-adic field) by means of twisted local gamma factors (the Local Converse Theorem). As applications, we prove that two irreducible generic cuspidal automorphic representations of ${\rm SO}_{2n+1}({\Bbb A})$ (where ${\Bbb A}$ is the ring of adeles of a number field) are equivalent if their local components are equivalent at almost all local places (the Rigidity Theorem); and prove the Local Langlands Reciprocity Conjecture for generic supercuspidal representations of ${\rm SO}_{2n+1}(k)$.