#### Vol. 14, No. 9, 2020

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

### Hengfei Lu

Vol. 14 (2020), No. 9, 2417–2480
##### 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 $\stackrel{̄}{\tau }$ of ${PGSp}_{4}\left(E\right)$.

##### Keywords
Langlands correspondence, see-saw diagrams, theta lift, quaternionic Hermitian groups, the Prasad conjecture
Primary: 22E50
Secondary: 11F27