11 problems

Let $F$ be a $p$-adic field, let $E/F$ be a degree-two extension, set $G={\rm GL}_n(E)$ and $H={\rm GL}_n(F)$, and let $\pi$ be an irreducible admissible representation of $G$. A r…

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…

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

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 $E/F$ 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 $E/F$ be a quadratic extension with nontrivial Galois element $\sigma$. Let $G$ be a reductive group over $F$, let $G(E)$ be its base change, and let $G^{\mathrm{op}}$ be the qu…

Prasad–Takloo-Bighash conjecture. Let $\pi$ be an irreducible admissible representation of $G$ with trivial central character. If $\pi$ is $H$-distinguished, then:

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 $F$ 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…

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