The Prasad conjectures for $\mathrm{GSp}_4$ and $\mathrm{PGSp}_4$

Abstract

We use the theta correspondence between ${GSp}_4\left(E\right)$ and $GO\left(V\right)$ to study the ${GSp}_4$-distinction problems over a quadratic extension $E∕F$ of nonarchimedean local fields of characteristic $0$. With a similar strategy, we investigate the distinction problem for the pair $\left({GSp}_4\left(E\right),{GSp}_{1,1}\left(F\right)\right)$, where ${GSp}_{1,1}$ is the unique inner form of ${GSp}_4$ defined over $F$. Then we verify the Prasad conjecture for a discrete series representation $\bar{\tau }$ of ${PGSp}_4\left(E\right)$.