41 problems

Let $\psi\in\Psi({}^{\vee}G^{\Gamma})$ be an Arthur parameter, let $\varphi_\psi$ be its associated Langlands parameter, and let $\Pi(G)_\varphi$ denote the corresponding $L$-packe…

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 $G$ be a connected reductive $p$-adic group and let $\psi:W_F”\to{}^LG$ be an Arthur parameter. Write $\phi_\psi$ for the associated Langlands parameter, and let…

Weak Arthur packet decomposition conjecture. There is a set $\mathcal T$ and an appropriate notion of Arthur packet such that

Let $\psi$ be an Arthur parameter, let $\varphi_{\psi}$ be its associated Langlands parameter, and let $\Pi(\varphi_{\psi})$ be the $L$-packet associated to $\varphi_{\psi}$. A rep…

Case 7 conjecture. Then

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…

In the setting of Theorem, let $G^{\Gamma}$ be an extended real reductive group, let $\psi$ be an Arthur parameter, and let the notation $\Pi^z(G^{\Gamma})^{mic}_{\psi}$ and…

Let $G=G(W_n)$ and let $\psi,\psi'\in\Psi^+(G_n)$ be local Arthur parameters with $\psi\geq_C\psi'$. Suppose that $\pi\in\Pi_\psi\cap\Pi_{\psi'}$. The closure-monotonicity conjectu…

Let $\pi$ be an irreducible admissible representation of good parity. The good-parity Arthur-type conjecture. The representation $\pi$ is of Arthur type if and only if it is unitar…

Let $(G,H)$ be a dual pair, let $\pi$ be a representation of $G$ of Arthur type, and let $\psi^{\max}(\pi)$ be the maximal Arthur parameter attached to $\pi$. Let $\alpha_0$ be the…

Let $G$ be a reductive group and let $\pi$ be an irreducible admissible representation of $G$. Define $\Psi(\pi)=\\{\psi\in\Psi^+(G)\mid \pi\in\Pi_\psi\\}$, let $\phi_\psi$ be the…

Let $G$ be a reductive group, let $\Psi^+(G)$ denote its local Arthur parameters, and let $\phi_\psi$ be the associated $L$-parameter. For a representation $\pi$, define … Let…

Let $G$ be a classical group, and let $\pi$ be an irreducible unitary representation of $G$ of good parity. A representation is of Arthur type when it belongs to an Arthur packet.…

Let $n$, $r$, $k_i$, and $m_i$ be as above, and let $\psi^\BR$ denote the corresponding parameter on the complex side. On the $p$-adic side, let $\psi^p$ and the Arthur–Vogan packe…

Let $\mathrm{G}$ be a connected reductive group over a non-Archimedean local field $k_v$, and let $G=\mathrm{G}(k_v)$. Assume that a theory of local Arthur packets for $G$ exists.…

Let $\pi$ be a representation of $G$ and let $\psi^{\max}(\pi)$ denote its maximal Arthur parameter. Let $d^\pm(\pi,\psi^{\max}(\pi))$ be the corresponding going-up or going-down i…

Let $\pi=\pi(\lambda,s,{{\mathcal O}}^\vee_{\lambda,s},{{\mathcal L}}^\vee)$ be an irreducible representation in the tempered $L$-packet…

Kim's conjecture. The parameter $\psi$ is associated with a distinguished orbit $\mathcal{O}$ of $G^{\vee}$, and

Let $G$ be a quasi-split reductive group, and let an Arthur packet mean the packet attached to an Arthur parameter for $G$. A member is locally generic with respect to a Whittaker…

Let $\psi_v$ be a local $A$-parameter, let $\Pi_{\psi_v}$ be its local $A$-packet, and let $\Pi_{\phi_{\psi_v}}\subset\Pi_{\psi_v}$ be the corresponding $L$-packet. Let $r(\pi)$ de…

The weak local Arthur packets conjecture. The set $\Pi_{\psi}^{\textrm{Weak}}$ is a union of local Arthur packets.

Adams's conjecture. The representation $theta_{-alpha}(pi)$ is contained in the A-packet parametrized by

Let $\psi$ be an Arthur parameter for a reductive group $\mathbf G$ over a nonarchimedean local field ${\mathsf k}$, and let $\mathcal O^{\vee}_{\psi}$ be the nilpotent $G^{\vee}$-…

Theta-packet Arthur multiplicity conjecture. There is a decomposition