15 problems

Let $G$ be a semisimple $\mathbb{Q}$-group admitting a reductive $\mathbb{Z}$-model. Let $\pi\in\Pi_{\mathrm{disc}}(G)$ and let $r:\widehat{G}\to\mathrm{SL}_{n,\mathbb{C}}$ be an a…

Fix a distinguished character $\overline{\chi}^0$. Let $\mathcal{A}_\epsilon(\overline{\operatorname{Sp}}_{2r}(\mathbb A))$ denote the relevant discrete-spectrum representation spa…

Arthur parameter conjecture. Every Langlands parameter $\sigma$ occurring in the canonical decomposition is of the form $\sigma_\psi$ for an elliptic Arthur parameter $\psi$. This…

Let $G$ be a semisimple algebraic group over ${\mathbb Q}$, let $L$ be the Levi subgroup associated with a $\theta$-stable parabolic subalgebra $\mathfrak{q}$, let $\tau$ be a unit…

Let $G$ be a quasi-split connected reductive group over $\mathbb Q$, let $N$ be the unipotent radical of a Borel subgroup, and fix a non-degenerate character $\psi_N$ of…

Let $F$ be a non-archimedean local field, let $\Delta\operatorname{SL}_2$ denote the diagonally embedded copy of $\operatorname{SL}_2$ in…

Let $F=k((t))$ be a local field, let $G(F)$ be the group of $F$-rational points, and let $\pi$ be a tempered irreducible admissible representation of $G(F)$. Let $AZ(\pi)$ be its i…

For a global Arthur parameter $\psi\in\widetilde{\Psi}_2(G_n)$, let $\widetilde{\Pi}_{\psi}(\epsilon_\psi)$ be its automorphic $L^2$-packet, and let $\underline{p}(\psi)$ be the pa…

Let $Y=H\backslash G$ be a strongly tempered spherical variety over a local field $F$. Let $G_Y^\vee$ be the dual group of $Y$, let…

Let $G$ be the reductive group under consideration, let $X$ be the underlying curve, and let $\check{G}$ be its Langlands dual group. Let … be a distinguished unipotent Arthur para…

Generic Summand Conjecture. There exists a cuspidal realization ${\mathcal {C}}_\pi$ with first occurrence index $\ell_0$ and a suitable $k$-rational orbit…

Bessel-period obstruction conjecture. The integer $\ell$, with $0\leq\ell\leq {\mathfrak {r}}_n$, is not the first occurrence index of $\pi$.

Strong generic Arthur packet conjecture. If $\Pi(\phi_{\psi})$ has a generic member, then $\Pi(\phi_{\psi})$ is a tempered $L$-packet; equivalently, all members of…

Weak generic Arthur packet conjecture. If $\Pi(\phi_{\psi})$ has a generic member, then the Langlands parameter $\phi_{\psi}$ is tempered.

Generalized Ramanujan conjecture. Each constituent $\pi_i$ is tempered.