The local converse theorem for $Mp_{2n}$ : the generic case

TL;DR

This paper proves the local converse theorem and stability of gamma factors for the metaplectic group using theta correspondence, and establishes a rigidity theorem for automorphic representations.

Contribution

It introduces new proofs of the local converse theorem and gamma factor stability for $ ext{Mp}_{2n}$, leveraging the local theta correspondence with $ ext{SO}_{2n+1}$.

Findings

Established the local converse theorem for $ ext{Mp}_{2n}$.

Proved stability of local gamma factors for $ ext{Mp}_{2n}$.

Proved the rigidity theorem for automorphic representations of $ ext{Mp}_{2n}$.

Abstract

In this paper, we establish the local converse theorem and the stability of local gamma factors for $\Mp_{2n}$ via the precise local theta correspondence between $\Mp_{2n}$ and $\SO_{2n+1}$ over local fields of characteristic not equal to 2. We also prove the rigidity theorem for irreducible generic cuspidal automorphic representations of $\Mp_{2n}$ over number fields.