5 problems

Let $G$ be a reductive group over a non-Archimedean field, let $\phi$ be an $L$-parameter, and let $\Pi_{\phi}^{\mathrm{ABV}}$ denote its ABV-packet. Let $\widehat{\phi}$ be the Py…

The refined Adams conjecture for ABV-packets. If $\pi\in\Pi_\phi^{\mathrm{ABV}}$, then: (1) for $\alpha\gg0$, $\theta_{-\alpha}^\pm(\pi)\in\Pi_{\phi_\alpha}^{\mathrm{ABV}}$; (2) if…

Let $G$ be any quasi-split reductive group over a $p$-adic field $F$, and let $\phi$ be a local Langlands parameter of $G$. An open parameter is a parameter whose orbit is open in…

Let $\mathrm G$ be a connected reductive group over $F$ with a quasi-split pure inner form, and let $G=\mathrm G(F)$. Let $\Phi(G)$ be the set of local $L$-parameters, let…

Let $G=GL_n$, and consider ABV-packets attached to orbits of Arthur type for $G$. Singleton conjecture. ABV-packets for orbits of Arthur type are singletons when $G=GL_n$. This is…