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

Abstract

Let $F$ be a $p$-adic field and $\pi$ be an irreducible smooth representation of ${\text{Sp}}_{2n}(F)$. 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 $\kappa \in {\text{GSp}}_{2n}(F)$. 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 ${\text{Sp}}_{2n}\times {\text{GL}}_m$ on $\psi$ to the general case.

arXiv commentThis paper has been withdrawn by the author since the result proved paper could be derived easily from known facts