57 problems

Let $G$ be a split classical group, and let $\pi$ be an isolated representation in the unitary dual of $G$. Automorphy conjecture. The representation $\pi$ is automorphic. The conj…

Let $F$ be a local field of characteristic zero, let $G$ be a classical group over $F$, and let $\pi\in\Pi_F(G)$ have a generic local $L$-parameter. For each symbol $\square$, let…

Let $k$ be a field of characteristic zero, let $I$ be the unit for the tensor product, and let $x$ be an object of dimension $n$ with $n$th exterior power $\Lambda^n(x)$. The speci…

Let $G$ be the $p$-adic classical group considered in the paper, and let $\pi$ be an irreducible supercuspidal representation of $G$. A representation is compactly induced if it is…

Characterization conjecture. When $J_+=\emptyset$, the map $j_\beta$ is characterized by property d), namely compatibility with the Lie algebra filtrations of $\mathfrak g$ and…

Let $V$ be a vector space with a bilinear form, and let $G:=\operatorname{Aut}(V,\langle\cdot,\cdot\rangle)$ be respectively a general linear, orthogonal, or symplectic group accor…

The refined Adams conjecture for ABV-packets. If $\pi\in\Pi_\phi^{\mathrm{ABV}}$, then: (1) for $\alpha\gg0$, $\theta_{-\alpha}^\pm(\pi)\in\Pi_{\phi_\alpha}^{\mathrm{ABV}}$; (2) if…

Let $(G,H)$ be a dual pair, let $\pi$ be a representation of $G$ of Arthur type, and let $\psi^{\max}(\pi)$ be the maximal Arthur parameter attached to $\pi$. Let $\alpha_0$ be the…

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 $F$ be a non-archimedean local field of characteristic $0$ and residual characteristic $p$, and let $\mathrm{G}(F)$ be the group of $F$-points of a split symplectic or special…

Derangement proportion conjecture. The following formulas hold:

Let $F$ be a non-Archimedean local field of characteristic zero, let $G_n=\mathrm{Sp}_{2n}(F)$ or $G_n=\mathrm{SO}_{2n+1}(F)$, and let $\psi\in\Psi^+_{\mathrm{unit}}(G_n)$ have dec…

Let $k$ be an algebraically closed field of characteristic zero, let $I$ be the unit for the tensor product, and let $x$ be an object of dimension $n$ equipped with an isomorphism…

Let $k$ be an algebraically closed field of characteristic zero, let $I$ be the unit for the tensor product, and let $x$ be a self-dual object of dimension $n$. Let…

Let $k$ be a field of characteristic zero, let $I$ be the unit for the tensor product, and let $x$ be a self-dual object of dimension $n$. Let $\epsilon\colon x\otimes x\to I$ be i…

Let $k$ be a field of characteristic zero. A 2-rig is the representation-theoretic structure in which the stated universal property is considered; let $x$ be an object, let $n$ be…

Let $e$ be a nilpotent element in a classical Lie algebra $\mathfrak{g}$, with partition $\lambda$, and let $\lambda^\intercal$ be its dual partition. Form $\lambda_{\mathrm{red}}$…

Let $d\geq 4$, let $q$ be a prime power, and let $\operatorname{SL}(d,q)$ be given by a generating set. Let $\xi$ denote an upper bound on the number of field operations for comput…

Let $G$ be one of $\mathrm{SU}(n)$, $\mathrm{SO}(n)$, $\mathrm{Spin}(n)$, or $\mathrm{Sp}(n)$, and let $\mathcal{A}\subseteq G$ be measurable with measure $\nu$. The diameter with…

Let $G$ be either ${\rm Sp}(V)$ or $\Omega(V)$, let $u$ be the representative in … J={j:bj=2}. … . The centralizer quotient order conjecture. One has … This conjecture specifies th…

Let $G$) be one of the classical groups under consideration, with standard representation $\mathrm{Std}$, and let $(\delta=0,\mu=\rho\times\phi)$ be rigid hypergeometric automorphi…

Maximal wave front partition conjecture. If $G_{n^*}=\operatorname{Sp}_{2n^*}$ or $\operatorname{O}_{2n^*}$, then

Let $G$ be $Sp(2n,\mathbb{R})$ or $SO^*(2n)$, and consider the class of unitary representations constructed in Section 8 of Vogan's work, including all weakly fair…

Classical compact-group order conjecture. The leading order of these critical-subcritical moments of moments is conjectured to be $\Omega(N\log N)$ for all three classical compact…

Let $\pi$ be an isolated representation in the unitary dual of a classical group. Isolated-representation Arthur-packet conjecture. The representation $\pi$ belongs to an Arthur pa…