91 problems

Let $G$ be a reductive group over a nonarchimedean local field $F$, let $\varphi$ be a discrete Langlands parameter, and let $\pi \in \Pi_{\varphi}^2(G,F)$ correspond to an irreduc…

Let $G$ be a quasi-split reductive group over a non-Archimedean local field $F$, let $G'$ be an inner form, and let enhanced L-parameters for $G'$ be defined as pairs…

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…

Depth-preservation conjecture. For every $\pi\in\Pi(\phi)$,

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

Let $G^{*}$ be a connected, reductive, quasi-split group over the local function field $F$, with finite central $F$-subgroup $Z$, and let $G$ be an inner form of $G^{*}$. Fix a Whi…

Cuspidal-support compatibility conjecture. Under the local Langlands correspondence, the map $\mathbf{Sc}$ corresponds to Bernstein's cuspidal support map for irreducible smooth…

Cuspidal LLC conjecture. Under the local Langlands correspondence, $\Phi_{\mathrm{cusp}}(G)$ corresponds to the set of supercuspidal irreducible smooth $G$-representations. This pr…

Let $F$ be a non-archimedean local field and let $n\geq 2$. Consider the group $\operatorname{GL}_n(F)$ and the corresponding Lubin–Tate spaces of dimension $n-1$, together with th…

Langlands's conjecture. Every elliptic $\lambda$ corresponds to a finite set $\Pi_{\lambda}$, called an $L$-packet, of cuspidal irreducible representations of $G(E)$.

Let $W_k$ be the Weil group of a nonarchimedean local field $k$, let ${\cal G}^0_{2n}(k)$ be the set of conjugacy classes of $2n$-dimensional admissible completely reducible multip…

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:

Stable center conjecture. The subspace $\mathcal{Z}^{st}(G)\subset\mathcal{Z}(G)$ is a unital subalgebra. This conjectural stable-center structure is related to a decomposition of…

Let $k$ be a non-Archimedean local field of characteristic zero, let $G$ be the $k$-points of a connected reductive algebraic group, and let $\Pi_u(G)$ be the set of equivalence cl…

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…

Let $G$ be a connected reductive $F$-group that splits over $F^t$, and let $\varphi:W'_F\to{}^LG$ be a Langlands parameter, where $W'_F=W_F\times\mathrm{SL}_2$. Let $\Pi(\varphi)$…

Let $G$ be a connected reductive $F$-group that splits over $F^t$, and let $W'_F=W_F\times\mathrm{SL}_2$. For an irreducible smooth representation $\pi$ of $G(F)$, let $(x,X)$ be a…

Internal parametrization conjecture. There exists a canonical surjection

Kaletha's internal parametrization conjecture. The packet on $(G',\iota,z)$ should be internally labeled by

Depth-preservation conjecture. The depths agree:

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…