92 problems

Let $G$ be the ambient reductive group, let $P$ be a parabolic subgroup of $G$ containing $B$, with Lie algebra $\mathfrak{p}$ and opposite parabolic $\overline{P}$. Let $M$ be a…

Genericity conjecture. A pure $L$-packet $\Pi^{\text{pure}}_\phi(G)$ is generic if and only if $\phi$ is an open parameter for $G$.

Let $(T,U)$ be a minimal elliptic pair, and let $\chi\colon \mathcal{T}^{+}(\mathcal{O}_k)\to\overline{\mathbb{Q}}_\ell^{\times}$ be a smooth character. Let $s_\chi\geq 0$ denote t…

Let $\mathbf{G}$ be a reductive group over the nonarchimedean field $k$, let $\mathbf{X}$ be a spherical $\mathbf{G}$-variety, let $K$ be a maximal compact subgroup of…

Let $E$ be the number field in the deformation problem, let $V$ be an absolutely irreducible crystalline $p$-adic representation with the stated self-duality and distinct local dat…

Let $X$ be the building of $G$, and let ${\mathscr{S}}$ be the Schneider–Stuhler $G$-equivariant sheaf associated with an irreducible supercuspidal representation. A basic elementa…

Let $(\pi,V)$ be an irreducible supercuspidal representation of $G$, and let ${\mathscr{S}}$ be the $G$-equivariant Schneider–Stuhler sheaf on the building $X$ associated with…

Let $F$ be a nonarchimedean local field, let $\boldsymbol{\mathrm{G}}$ be a connected reductive group defined over $F$, and let $G=\boldsymbol{\mathrm{G}}(F)$ with center $Z$. Let…

Gross's conjecture. The following assertions hold: $\pi^{K_{n,m}}=0$ for $0\leq m<c_\pi$ and $\dim_\mathbb{C}\pi^{K_{n,c_\pi}}=1$; the action of $J_{n,c_\pi}/K_{n,c_\pi}$ on…

Let $\mathcal{L},\mathcal{L}'\in\mathbb{Q}_p$ be admissible, let $k$ be the weight, and let $C_{p,k}$ be the constant appearing in the conjecture. Let…

Factorization conjecture. The composite $\iota_2\circ\iota_1$ factors through $\iota_3$.

Unique embedding conjecture. The embedding

Generalized injectivity conjecture. Let $\pi\in\mathrm{Irr}_{\rho}$ and let $\mathfrak n\in\mathrm{Mult}_{\rho}$ be minimal to $\pi$. Set

Let $D$ be the filtered $\varphi$-module under consideration, let $\sigma$ range over the relevant embeddings, and let $I$ range over subsets that are very critical for $\sigma$. L…

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

Aubert–Zelevinsky duality conjecture. If $X=X(q^{\frac{1}{2}{\it h}},\mathcal{S},\mathcal{L})$ is an irreducible unipotent representation in…

Let $G$ be a reductive group over a $p$-adic field $F$, let $x$ be a point in its Bruhat–Tits building, and let $K_\ell:=K_{G,x,\ell}$ be the $\ell$th Moy–Prasad subgroup at $x$. W…

Assume that $G$ is quasi-split over $F$, equipped with a pinning $(B,T,e)$, and fix a non-trivial additive character $\psi:F\to {\mathbb Z}_\ell[\mu_{p^\infty}]^\times$. Let…

Let $e$ be a finite sum of primitive idempotents in the center $Z_n$ of $\operatorname{Rep}_{\mathcal{O}}(G_n)$, let $\mathcal{P}_{\lambda}$ be the associated projective object, le…

Let $\Delta=\{\alpha_1,\dots,\alpha_{n-1}\}$, and let $V$ be an object of $\operatorname{Rep}^{\operatorname{ord}}_h(G_{\mathbb{Q}_p,\Delta}\times\mathbb{Q}_p^\times)$. For each…

Explicit splitting conjecture. For $\ell\in\{2,\dots,n-2\}$, the morphism of complexes

Let $X$ be as in the paper, let $n\geq 2$, let $G(X)$ be a reductive group, and let $\Pi_X[\log(1+T)^{1/n}]$ be the generalized Robba ring in the imperfect setting. Let…

Let $X$ be a finite extension of a local nonarchimedean field, let $G(X)$ be a $z$-adic reductive group, and let $\mathrm{Gal}_{X,2}$ denote the two-fold covering of the Galois gro…

Let $K$ be a finite extension of $\mathbb{Q}_p$ or $\mathbb{F}_q((z))$ with $\ell\neq p$ and $p>2$. The source considers motivic Schur-irreducible sheaves in the derived categories…

Let $b\in B(G)$ be basic, let $\mu_\bullet$ be a finite collection of cocharacters, and let $\rho$ be an irreducible admissible representation of $G_b(\bbQ_p)$ whose parameter…