5 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…

Let $\pi\in\Pi(G_n)$ be a tempered representation, let $m^{{\mathrm{up}},\alpha}(\pi)$ be the odd integer determined by its first occurrence in the going-up tower, and let…

Let $m=n$ or $n+1$, let $\pi$ be square-integrable, and let $\sigma=\Theta_{V,W,\psi_\tau}(\pi)$ be irreducible and square-integrable. Under these conditions, the last integral in…

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…