164 problems

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…

Fix a quasi-split reductive group $G$ over a local field $E$, and let $W_E$ be its Weil group. Let $\check{G}$ be the Langlands dual group, and let…

Let $G$ be a connected reductive $F$-split group. For a unipotent element $u otin G^\text{?}$, a compact pair $(s,h) otin \text{?}$, and a maximal compact open subgroup $K$ of $G$, l…

Let $\theta$ be an $m$-realization of $\mathfrak{t}$ for $({\rm G},h)$ with parametrization $(({\rm V},h),\varphi,\Lambda,\beta)$. Let $\mathcal{C}$ be a chamber in the building of…

Let $G_n$ be the classical group under consideration. Let $\Pi_{\overline{A}}^{\lim}(G_n)$ and $\Pi_{\overline{A}}^{\lim'}(G_n)$ denote the sets defined in the preceding discussion…

Let $G$ be a classical group, and let $\pi$ be an irreducible unitary representation of $G$ of good parity. A representation is of Arthur type when it belongs to an Arthur packet.…

Let $F$ be a non-Archimedean local field of characteristic zero, and let $G$ be a general connected reductive group defined over $F$. Let $\Pi_A(G)$ denote the set of Arthur repres…

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…

Let $H$ be a $p$-adic reductive group, let $\lambda^{\mathbb{Q}_p}$ be an infinitesimal character, and let $\xi=(C,\mathcal{V})$ and $\xi'$ be parameters indexing the corresponding…

Let $H$ be the reductive group and let $\widehat{P}\subseteq\widehat{H}$ be the parabolic data defining the Lusztig induction functor … On the level of Grothendieck groups, the dua…

Assume that $\rho$ factors through $\widehat{G}(\mathbb{C})\rtimes\operatorname{Gal}(E/F)$ for a finite Galois extension $E/F$, and let $M_\rho$ be Ngô's reductive monoid attached…

Muić's conjecture. The big theta lift $\Theta(\pi)$ to $H(V_m)$ is irreducible, if non-zero.

Schneider–Ollivier's surjectivity conjecture. The canonical morphism $\mathbb{X}^*\to\mathcal{H}$ is a surjection.