On the dependence of the local Rankin-Selberg gamma factors of $\textrm{Sp}_{2n}\times \textrm{GL}_m$ on $\psi$

TL;DR

This paper proves a symmetry property of certain representations of symplectic groups over p-adic fields and extends the known dependence of local Rankin-Selberg gamma factors on the additive character to more general cases.

Contribution

It establishes the equivalence of specific generic representations under conjugation and extends the dependence relation of gamma factors on the additive character.

Findings

Representation equivalence under conjugation for generic representations.

Extension of gamma factor dependence on additive character.

Confirmation of conjectures based on local Langlands and Gan-Gross-Prasad.

Abstract

Let $F$ be a $p$-adic field and $\pi$ be an irreducible smooth representation of $\textrm{Sp}_{2n}(F)$. In this paper, we show that if $\pi$ and $\pi^\kappa$ are both generic for a common generic character of the maximal unipotent of a fixed Borel, then $\pi\cong \pi^\kappa$, where $\pi^\kappa$ is the representation induced by the conjugation action of an element $\kappa\in \textrm{GSp}_{2n}(F)$. This result is a consequence of the standard local Langlands conjecture and local Gan-Gross-Prasad conjecture. As a consequence, we extend the dependence relation of the local Rankin-Selberg gamma factors for $\textrm{Sp}_{2n}\times \textrm{GL}_m$ on $\psi$ to the general case.