43 problems

Smooth transfer conjecture. For any relative endoscopic datum $(\xi,\alpha,\beta)$ and any $f\in C_c^\infty(\mathfrak g_1)$, there exists…

Langlands–Shelstad transfer conjecture. For every $G_v'\in{{\mathscr E}}(G_v)$, this transfer function exists and the resulting map sends ${{\mathscr H}}(G_v,\zeta_v)$ continuously…

Let $\mathbf{G}$ and $\mathbf{H}$ be reductive groups, set $G=\mathbf{G}(F)$ and $H=\mathbf{H}(F)$, and let $H'$ be the subset of $\mathbf{G}$-regular elements of $H$. For a charac…

Let $(l,m)$ be regular. The endoscopic contribution $e_{\rm endo}({\mathcal A}_2,\mathbb V_{l,m})$ is defined using the cohomology of the moduli space ${\mathcal A}_2$ of principal…

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 $G$ be a reductive group over the local field $F$, let $H$ be an endoscopic group, and let $\gamma\in H(F)$ be strongly $G$-regular semisimple. Write $\Lambda_{G,H}(\gamma)$ fo…

The fundamental lemma. The orbital integrals satisfy

Let local components of automorphic representations be representations of real or $p$-adic reductive groups. Arthur's endoscopic character relation conjecture. These local componen…

Let $G$ be the relevant reductive group and let $\operatorname{LLC}$ denote the local Langlands correspondence defined in the theorem referenced by the source. Let an endoscopic da…

Let $G$ be the reductive group under consideration, let $F$ be its local field, and let $\iota:\overline{\mathbb{Q}}_{\ell}\overset{\sim}{\to}\mathbb{C}$ be a field isomorphism. Le…

Endoscopic equivalence conjecture. There exists an endoscopic equivalence as in Theorem $$ without any hypothesis on $\ell$.

Let $F$ be a non-archimedean local field, let $G$ be a connected reductive $F$-group, and let $\Pi_{\varphi}(G)$ be a discrete series $L$-packet of irreducible smooth representatio…

Let $G$ be a quasi-split connected reductive group over a local or global field $F$ of characteristic zero. The conjectural Arthur–Langlands data consist of global parameters…

Local Langlands conjecture. There is a finite set $\Pi_\phi(G)$ of irreducible representations of $G(F)$ and an injective map

Let $\X$ be a symmetric variety and let the construction of an endoscopic variety associated to $\X$ be as in the paper. Endoscopic-variety conjecture. This construction should pla…

The epsilon-character conjecture. The function $\omega_{\phi',\rho_X}$ is well defined, independent of the extended endoscopic triple and the lifting, descends to a function on…

Let $(G,H)$ be a strongly tempered spherical pair that is the Whittaker induction of $(G_0,H_0,\xi)$, with $G$ admitting a quasi-split pure inner form $G_{qs}$. Assume that…

Let $\phi:L_F\to{}^LG$ be a tempered Langlands parameter, let $\tilde s\in\tilde S_\phi^{[z]}$ be semisimple, and let $\mathfrak e$ be the associated endoscopic datum. Choose a $z$…

Let $G$ be a reductive group with cover $G(F)_{x_G}$, let $(\dot s,\mathcal H)$ be a refined endoscopic datum for $G$, and let $H(F)_{x_H}$ be the corresponding endoscopic cover. L…

Let $G$ be a reductive group, let $\mu$ be a cocharacter with reflex field $E_{\mu}$, and let $(H,\mathcal{H},s,\eta)$ be an endoscopic datum such that an $L$-parameter $\phi$ fact…

Let $\tilde{G}=G\rtimes\theta$ be a twisted space with an involution $\theta$ on a reductive group $G$, and let $H$ be an endoscopic group of $\tilde{G}$. Assume that $G$ and $H$ a…

Let $G$ be a quasi-split connected reductive $F$-group, let $\varphi:L_F\to{}^LG$ be a Langlands parameter, and let $\Pi_\varphi$ be the compound $L$-packet of representations of a…

Weighted Langlands–Shelstad transfer conjecture. For each $G$ and $M$, there is a function $\hat{S}^{{\tilde G}'}_{{\tilde M}'}(\delta',f')$ on … such that

For $\theta \in \Sigma_0$ and a semisimple element $s \in \bar{S}_{\phi}^{\theta}$, let $(G',\phi') \rightarrow (\phi,s)$ be the corresponding endoscopic relation, and let…

Let $(G,b,\mu)$ be the local Shimura data appearing in the paper, let $\mathcal{H}^{\mathfrak{e}}$ be an endoscopic datum, and let $\operatorname{Mant}_{G,b,\mu}$ and…