On the Local Langlands Functorial Transfer from $\text{SO}(5)$ to $\text{GL}(4)$

Abstract

Let $\text{F}$ be a non-archimedean local field of characteristic zero. We study the local Langlands functorial transfer from $\text{SO}(5, \text{F})$ to $\text{GL}(4, \text{F})$ for several families of supercuspidal representations of $\text{GL}(4, \text{F})$. In doing so, we give equivalent conditions for such representations to be functorial transfers from $\text{SO}(5, \text{F})$, expressed in terms of the Bushnell--Kutzko construction of supercuspidal representations, by studying poles of local exterior square $L$-functions and the existence of non-zero local Shalika models.