24 problems

Let $E/F$ be a quadratic extension, let $G$ be a reductive group over $F$, and let $N_0$ be the unipotent subgroup associated with a fixed Whittaker datum. Let $\pi$ be an irreduci…

Let $F/F_{0}$ be a quadratic extension of $p$-adic fields, let $G=\mathrm{GL}_{n}(F)$), and let $\sigma$ be the non-trivial automorphism of $F/F_{0}$. Let $\tau$ be a unitary invol…

Lapid–Prasad conjecture. If $\pi$ is $H$-distinguished, then the $L$-packet of $\pi$ is invariant under the functor

Let $F$ be a $p$-adic field, let $E/F$ be a degree-two extension with nontrivial Galois element $\sigma$, set $G={\rm GL}_n(E)$ and $H={\rm GL}_n(F)$, and let $\omega_{E/F}$ be the…

Main conjecture. An irreducible smooth representation $\theta$ of $\operatorname{G}_n$ with $n>2$ is $\operatorname{H}_{1,n-1}$-distinguished if and only if $\theta$ is either…

Let $\operatorname{E}/\operatorname{F}$ be the setting of distinguished representations, and let $\pi$ be a supercuspidal representation of $\operatorname{GL}(n,\operatorname{E})$…

Let $G_n$ be the relevant general linear group, let $G_n'$ be its distinguished subgroup, and let $pi$ be an irreducible $G_n'$-distinguished representation of $G_n$ with depth $d$…

Let $E/k$ be the quadratic extension implicit in $\operatorname{Res}_{E/k}(\operatorname{GL}_n)$, set $\operatorname{G}=\operatorname{Res}_{E/k}(\operatorname{GL}_n)$ and…

Distinguished-lift conjecture. The following assertions are equivalent: $\pi$ is distinguished; $\pi$ has a distinguished lift to $\overline{\mathbb{Q}}_\ell$; and the three condit…

Let $E/F$ be a quadratic extension of local fields, let $G$ be a reductive group split over $E$, and let $\theta$ denote the nontrivial Galois conjugation on $G(E)$. A representati…

Let $F$ be a non-Archimedean local field of characteristic $0$, let $E/F$ be quadratic, and set $G=\operatorname{Res}_{E/F}\mathrm{GL}_{2m}$, so that $G(F)=\mathrm{GL}_{2m}(E)$. Le…

Let $E/F$ be a quadratic extension of number fields, let $D$ be a quaternion algebra over $F$ into which $E$ embeds, and let $\pi$ be a distinguished cuspidal automorphic represent…

Relative endoscopy conjecture. A theory analogous to twisted endoscopy should relate distinguished representations on a classical group to representations on $\mathrm{GL}_n$ that a…

Let $E/F$ be a quadratic extension of non-archimedean local fields. Let $\mathrm{U}_n$ be the quasi-split unitary group associated with $E/F$, and let the stable and unstable base…

Let $E/F$ be a quadratic extension of non-archimedean local fields, let $\tau$ be the nontrivial element of $\operatorname{Gal}(E/F)$, and let $\chi_{-1}$ be a character of…

Let $G$ be a reductive algebraic group over $\mathbb{R}$, and let $\pi$ be an irreducible representation of $G(\mathbb{C})$. Complex-real distinction conjecture. If…

Let $T$ be a torus over a local field $F$, let $E/F$ be the quadratic extension under consideration, and let $T^{\rm op}$ be the corresponding auxiliary torus. Torus base-change co…

Let $G$ be a quasi-split group over a local field $F$, let $E/F$ be a separable quadratic extension, and let $\pi$ be an irreducible admissible representation of $G(E)$. Let…

Let $F$ be the underlying field, and let $\tau$ be an irreducible generic representation of $\mathrm{GL}_n(F)$. A representation is distinguished when it has the distinguishedness…

Functorial lift conjecture. The $GL(n,F)$-distinguished unitary representations of $GL(n,K)$ should be precisely the unitary part of the image of a functorial lift, in the Langland…

Let $K/F$ be a quadratic extension, and let $\sigma$ denote the nontrivial element of its Galois group. Let $m$ be a positive integer, and let $\rho$ be a generic representation of…

Let $F$subset K be the quadratic extension of local fields, let $G_n(K)=GL(n,K)$, and let $\sigma$ denote the nontrivial element of $\operatorname{Gal}(K/F)$. For a representation…

Distinction conjecture. The representation $\pi$ is distinguished if and only if, after reordering the $\Delta_i$, there is an integer $r$ between $1$ and $t/2$ such that

Jacquet's conjecture. For representations $c0$ whose central character is trivial on $F^*$, the condition