5 problems

Let $G$ be a reductive group with Langlands parameter $\varphi$ satisfying $I_F\subseteq\ker\varphi$ and … where $\lambda$ is dominant integral in the fundamental positive chamber.…

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…

Prasad's conjecture. The representation $\pi$ is $H$-distinguished if and only if both the restriction of $\phi$ to $W_E$ is trivial and $\pi$ is generic. Under the first condition…

Connectedness conjecture. For any pair $(\phi,\alpha)$, the $\mathcal{O}_{K_e}[1/p]$-scheme ${\underline Z}^{1}(W_F^0,{\hat G})_{\phi,\alpha}$ is connected and remains connected af…