166 problems

Let $\mathrm{D}$ be a quaternion division algebra over a non-Archimedean local field $\mathrm{F}$ of characteristic zero, let $\mathrm{G}_n=\mathrm{GL}_n(\mathrm{D})$, and let…

Let $F$ be a local non-Archimedean field and let $\mathfrak m$ be any multisegment. If $\Lambda(\mathfrak m)$ denotes the associated RSK-standard representation of…

Let $\lambda$ be a Hodge type, viewed as an element of the character lattice of $G$, and let $\tau$ be a tame inertial type. Let $V(\lambda-\eta_{\mathbb{Q}_p})$ denote the restric…

Let $G=Sp_4(L)$, where $L$ is as in the paper. For $bin G(L)$, let … be the $\sigma$-centralizer of $b$, and let $\operatorname{def}(b)$ be the difference between the $F$-rank of…

Self-similarity conjecture. Assume that $n\geq 2$ and $p\geq d(n^2-1)$. Then no open subgroup of $SL_1^{ne}(D)$ is self-similar.

Let $F$) be a non-Archimedean local field of characteristic zero, let $\mathrm{G}$ be a connected reductive group over $F$, and set $G=\mathrm{G}(F)$. Let $\Pi_u(G)$ denote the uni…

Prasad's conjecture. The representation $\pi$ is $H$-distinguished if and only if both the restriction of $\phi$ to $W_E$ is trivial and $\pi$ is generic. Under the first condition…

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 $F$ be a non-Archimedean local field, let $G$ be the reductive group under consideration, and let $f^{EP}$ be the Euler–Poincaré function with respect to a Haar measure $dg$ on…

Refined local Langlands correspondence for $p$-adic quasi-split groups. The following should hold: for every $L$-parameter $\varphi$, there is a bijection

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

Let $\Pi$ be a cuspidal support, let $\mathcal{O}(\Pi)$ denote the corresponding Zelevinsky line, and let $\mathbf{a}$ and $\mathbf{b}$ be multisegments belonging to the same…

Binary-operation conjecture. $\pi(C_1*C_2)$ is a subrepresentation of $\pi(C_1)\times\pi(C_2)$. In particular, if $\pi(C_1)$ or $\pi(C_2)$ is $\square$-irreducible, then

Endo-parameter compatibility conjecture. The diagram relating $\operatorname{Irr}(G)$, $LP_G(W_F)$, $\mathcal{E}_{par}(F)$, and the representations of $P_F$ is commutative.

Let $\mathbf G=\mathrm{GL}(k\infty,\mathbb{Q}_p)$, and let $\mathbf K$ be its compact subgroup. For each positive integer $l$ and each $r\in\mathrm{Sp}(2kl,\mathbb{Q}_p)$, let…

Extension-group conjecture. For $a,b\in Q$, one has

Let $\mathfrak{o}$ be a discrete valuation ring with maximal ideal $\mathfrak{p}$ and residue-field cardinality $q=|\mathfrak{o}/\mathfrak{p}|$. For positive integers $\ell$ and…

Let $G$ be a compact $p$-adic analytic group. Write $\Omega_G$ for its completed group algebra over the coefficient ring used in the paper. Call $G$ dihedral-free if every orbital…

Let $G$ be one of the groups $SL_2$, $SL_3$, or $Sp_4$. Let $b=\epsilon^\nu$, and let $\ell(b)$ denote the length of the translation in $\widetilde W$ determined by $\nu$. Eventual…

Let $G$ be a connected reductive group over $F$, let $b\in G(L)$, and let $\mu$ be a dominant coweight such that the affine Deligne–Lusztig variety $X_\mu(b)$ is non-empty. Write…

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…

Let $G$ be a compact $p$-adic analytic group, let $H$ be a closed normal subgroup of $G$, and write $\Omega_{G,H}$ and $\Lambda_{G,H}$ for the corresponding Iwasawa algebras. For…

Multiplicity one conjecture. Multiplicity one holds for restriction of irreducible admissible representations of $\widetilde{{\mathbf{G}}}(k)$ to ${\mathbf{G}}(k)$.

Complete type-edge-pair conjecture. The union over all type-edge pairs equals the solution set:

Let $G$ be a quasi-split reductive group over $E$, let $\check{G}$ be its Langlands dual group, and assume $\ell\nmid\lvert\pi_0(Z(G))\rvert$. Let $W_E$ be the Weil group and let…