The Prasad conjectures for $\mathrm{GSp_4}$ and $\mathrm{PGSp_4}$

Abstract

In this paper, we use the theta correspondence between $\mathrm{GSp_4}$ and $\mathrm{GO(V)}$ to discuss the $\mathrm{GSp_4}$-distinction problems over a quadratic field extension $E/F.$ With a similar strategy, we study the period for the pair $(\mathrm{GSp_4(E)},\mathrm{GSp_{1,1}(F)}),$ where $\mathrm{GSp_{1,1}}$ is the unique inner form of $\mathrm{GSp_4}.$ Then we verify the Prasad conjecture for $\mathrm{PGSp_4(E)}$.