207 problems

Let $F$ be a local field, let $G$ be a reductive group over $F$, and let $WD_F$ be its Weil–Deligne group. Write ${}^LG=\hat{G}(\mathbb{C})\rtimes W_F$ for the $L$-group, and let…

Let $(\mathrm{G},b,\mu)$ be a PEL-type Rapoport–Zink datum, with $\mathrm{G}$ a connected reductive group over $\mathbb{Q}_p$, $\mu$ a minuscule cocharacter, and…

Let $G$ be a reductive group over the local field $E$, let $W_E$ be its Weil group, and let $\mathcal Z^{\mathrm{spec}}(G,\mathbb Q)$ be the natural rational form of the spectral B…

Let $K$ be a finite extension of $\mathbb{Q}_p$. Write $\mathcal{D}(\ast/GL_n(K)^{\la})$ for the stable $\infty$-category of locally analytic representations of $GL_n(K)$, let…

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…

Let $G$ be a reductive group, let $\Out_F(G)$ be its group of $F$-rational outer automorphisms, let $\theta\in\Out_F(G)$, and let $\mathfrak w$ be a Whittaker datum. Outer-automorp…

Let $T\subset G$ be a maximal torus split over $F$, with root datum $(X(T),\Phi(T),X^{\vee}(T),\Phi^{\vee}(T))$, and let $G^\sphat$ be the dual group. Define … and let…

Let $G$ be a connected reductive group over a local field $F$, let $\rho:{}^LG\to\operatorname{GL}_{V_\rho}(\mathbb{C})$ be a representation such that, for an isomorphism…

Let $G^*$ be an $F$-quasisplit classical group, let $\varphi$ be a local $L$-parameter, let $\Pi_\varphi[G^*]$ be the associated local Vogan packet, and let $\mathcal{S}_\varphi$ b…

Let $(G,H,\xi)$ be a Gross–Prasad triple attached to an admissible pair $(W,V)$ over $\mathbb{R}$. Let $m(\pi)=\dim \operatorname{Hom}_{H(\mathbb{R})}(\pi,\xi)$ for an irreducible…

Let $F/\mathbb{Q}_p$ be a finite extension with $p\neq 2$, let $G$ be a connected semisimple linear algebraic group split over $F$, and let $\mathcal{D}_F(G)$ denote the $G^\sphat$…

Let $G$ be a quasisplit reductive group over $E$, let $\Psi_G:\mathcal Z^{\mathrm{spec}}(G,\Lambda)\to\mathcal Z(G(E),\Lambda)$ be the map from the spectral to the Bernstein center…

Let ${\bf G}$ be an $F$-split connected reductive algebraic group defined over $F$, let ${\bf T}$ be a maximal $F$-split torus, and let $\xi$ and $\zeta$ be the parameters used to…

Let $G'$ be the relevant quasi-split unitary group, let $\phi'$ be an $L$-parameter of $G'$, and let $A_M$ be the finite group attached to the parameter as in the surrounding const…

Let $L'/L$ be a finite Galois extension of $p$-adic fields and let $K$ be a coefficient field. Fix an integer $d\geq 1$, an object $(r,N,V)$ of $\mathrm{WD}_{L'/L}$ of dimension…

Let $F$ be a local $p$-adic field and let $\mathbf{H}$ be a reductive group over $F$. Write $W_F$ for the Weil group of $F$ and $L_F$ for the corresponding Langlands group, and let…

Berenstein–Kazhdan's conjecture. For every algebraic $\ell$-dimensional representation $\rho$ of $G^\vee$, there exists a $W$-equivariant algebro-geometric distribution…

Let $F$ be a CM number field, let $\pi$ be a cuspidal, regular algebraic automorphic representation of $\operatorname{GL}_n(\mathbf{A}_F)$ of weight $\lambda$, and let…

Categorical local Langlands conjecture. There is an equivalence

Categorical local Langlands correspondence. For each standard Levi subgroup $M$ of $G$, there is a fully faithful functor

Let $G$ be a reductive group over a non-Archimedean field, let $\phi$ be an $L$-parameter, and let $\Pi_{\phi}^{\mathrm{ABV}}$ denote its ABV-packet. Let $\widehat{\phi}$ be the Py…

Let $\Phi_{\lambda}(G)$ be the set of $L$-parameters with fixed infinitesimal character, identified with the corresponding orbits in a Vogan variety. The geometric closure order on…

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 $H$ be a $p$-adic symplectic group or split special orthogonal group, and in the special even orthogonal case let $H^+$ be the corresponding full even orthogonal group. Let…