4 problems

Let $m,n\in\mathbb{Z}_{\geq 0}$, let $G_n=\operatorname{Sp}_{2n}(F)$ and $H_m^\pm=\operatorname{O}_{2m}^\pm(F)$ with $m>n$, and set $\alpha=2m-2n-1$. Let $\Pi(G_n)$ and…

Theta-lift transfer conjecture. If such a parameter $\phi'$ exists, then

Local theta correspondence conjecture. If $s\in S_\phi^+$ and $s'\in S_{\phi'}^+$ are associated via $\xi$, and $\pi\in\Pi_\phi(G(V))$, then $\theta_\psi(\pi,W)$ has $L$-parameter…

Howe duality conjecture. The set $\mathcal R_{\omega_{U,V}}(U,V)$ is the graph of a bijection between $\mathcal R_{\omega_{U,V}}(U)$ and $\mathcal R_{\omega_{U,V}}(V)$.