52 problems

Endo-parameter compatibility conjecture. The diagram relating $\operatorname{Irr}(G)$, $LP_G(W_F)$, $\mathcal{E}_{par}(F)$, and the representations of $P_F$ is commutative.

Let $F$ be a local field of characteristic zero, let $G$ be a classical group over $F$, and let $\pi\in\Pi_F(G)$ have a generic local $L$-parameter. For each symbol $\square$, let…

Let $G$ be a connected, quasi-split reductive group over $F$, let $(G',\xi,z)$ be a rigid inner twist of $G$, and let $\Phi_\mathrm{e}(G;[z])$ denote the set of $\widehat{G}$-conju…

Let $H$ be a split classical group over $F$, let $u$ denote the nilpotent orbit of the image of … under the derivative of the relevant $\mathrm{SL}_2(\mathbb{C})$-factor, and let…

Let $G$ be a connected reductive algebraic group over a locally compact non-Archimedean local field $F$. Let $\lambda:W_F\to{}^LG$ be an infinitesimal parameter. For geometric para…

Depth-$r$ stable-center conjecture. The image of $\xi^r$ is contained in the depth-$r$ stable center:

Let $G$ be a connected reductive group over $F$ that splits over a tame extension. Let $\pi$ be a depth-zero irreducible representation of $G(F)$, and let…

Let $G$ be a connected reductive group over $F$ that splits over $F^t$, let $\varphi$ be a positive-depth Langlands parameter for $G(F)$, and let $\Pi(\varphi)$ be its associated…

Internal parametrization conjecture. There exists a canonical surjection

Let $G$ be the reductive group under consideration, let $\operatorname{Temp}(G)$ be its tempered dual, and let $\pi\in\operatorname{Temp}(G)$. Let $\rho_{\pi,\varpi}$ be the repres…

Let $G$ be the underlying group, let $\textnormal{\textsf{Basic}}_G$ be the set of basic data, and for each $\beta\in\textnormal{\textsf{Basic}}_G$ let $\widetilde G_\beta$ be the…

Let $F$ be the local field and let $\widetilde G$ be the cover under consideration, with associated dual group data $\widetilde H$. Write $\Pi(\widetilde G)$ for the relevant set o…

Let $\bb G$ be a connected reductive group with Shimura datum $(\bb G,\bb X)$, let $\msf G$ be its relevant local base change, let $\mdc K^p$ be the prime-to-$p$ level, and let…

Let $\mathrm G$ be a connected reductive group and let $G=\mathrm G(F)$. Assume the local Langlands correspondence for $G$. For a local $L$-parameter $\phi$, let $\Pi_\phi$ be its…

Let $\mathrm G$ be a connected reductive group and let $G=\mathrm G(F)$. Assume the local Langlands correspondence for $G$. For a local $L$-parameter $\phi$, let $\Pi_\phi$ be its…

Refined local Langlands correspondence for $p$-adic quasi-split groups. The following should hold: for every $L$-parameter $\varphi$, there is a bijection

Let $G$ and $H$ be the groups occurring in the Local Langlands correspondence, let $\boldsymbol{Z}[[\sigma]$ be the group ring of…

Functoriality conjecture. The representation

Rigid refined local Langlands conjecture. There is a bijection, depending on $\mathfrak{w}$,

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

Hiraga--Ichino--Ikeda's formal degree conjecture. Suppose that $\pi\in\Pi(\mathbf{G})$ is a discrete series representation whose $L$-parameter is $\phi$. Then

Adler–Mishra–Stevens conjecture. The local Langlands correspondence for $L$ induces a bijection

Let $G$ be a split connected reductive group over a nonarchimedean local field $F$, let ${}^LG$ be its $L$-group with connected component $4\widehat{G}4$, and let…

Aubert–Moussaoui–Solleveld's Bernstein-block conjecture. The local Langlands correspondence for $L$ induces a bijection

Let $G$ be a reductive group, let ${\mathcal C}(G^\vee)_{\mathsf{ss}}$ be the set of semisimple conjugacy classes in the dual group, and for…