56 problems

Zassenhaus conjecture. If $u\in\mathcal{U}(\mathbb ZG)$ is a unit of finite order, then $u$ is conjugate within $\mathbb QG$ to $\pm g$ for some $g\in G$.

Let $k$ be one of the four levels … The vertex algebra under consideration is the exceptional W-algebra associated with the subregular nilpotent element $f_{subreg}$ of…

Let $G$ be a real semisimple group and $H\subset G$ a real semisimple subgroup, with $K^H=H\cap K$ maximal compact in $H$ and $(\mathfrak a^H_0)^+\subset\mathfrak a_0^+$. Let…

Let $\mathcal U$ be the relevant set of rigid orbits, and suppose that $G(1)$ is not $O(l)$ for any $l\geq2$. Unipotent-exhaustivity conjecture. The set…

Polycyclic cyclic escape conjecture. Every infinite finitely generated polycyclic group has the cyclic escape property.

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 $\rho_i:G_i\to\mathbf{U}(H)$ be faithful unirreps of connected simply connected Lie groups in a separable Hilbert space, with corresponding Lie algebras $\mathfrak{g}_i$. Let…

Let $\Gamma$ be a center-free irreducible higher-rank lattice in $G$, and let $V$ be a unitary representation of $\Gamma$ whose action does not extend to a $G$-action on any non-tr…

Let $\Gamma$ be a center-free irreducible higher-rank lattice in $G$, and let $V$ be a unitary representation of $\Gamma$. Assume that the $\Gamma$-action on $V$ does not extend to…

Let $G$ be a Lie group with Lie algebra ${\mathfrak g}$, let $h\in{\mathfrak g}$ be an Euler element, and let $G_{\tau_h}$ denote the subgroup associated with the involution determ…

Let $G$ be a reductive group with Langlands parameter $\varphi$ satisfying $I_F\subseteq\ker\varphi$ and … where $\lambda$ is dominant integral in the fundamental positive chamber.…

Uniform finiteness conjecture. One has

Ramond unitarity conjecture. The set of unitary highest weight modules over $W^k_{\min}(\mathfrak g)$ is the union of

Let $W^k_{\min}(\mathfrak g)$ be the minimal $W$-algebra, let $P^+_k$ denote the relevant set of dominant integral weights, and let $A(k,\nu)$ be the lower bound for the conformal…

Bekka–Valette conjecture. There exist unitary representations of $\Gamma$ that are not weakly contained in any representation restricted from $G$.

Let $G_{\mathbb{R}}$ be a real reductive group. Let $\Pi_u(G_{\mathbb{R}})$ be the set of equivalence classes of irreducible unitary $G_{\mathbb{R}}$-representations, and let…

Tadić's independence of unitarizability conjecture.

Tadić's preservation of unitarizability conjecture. If $\pi\in\Pi(G_n)$ is weakly real, then $\pi$ is unitary if and only if $X_{\rho_i}(\pi)$ is unitary for every $i=1,\ldots,r$.

Tadić's critical-type conjecture.

Good-parity Arthur-type conjecture. For a classical group $G$,

Unitary-dual exhaustion conjecture.

Vogan's strengthened FPP conjecture. Suppose $\Lambda$ does not lie inside FPP. Then $\pi$ is not unitary up to level $\mathfrak{p}$.

Vogan's FPP conjecture. If $\langle \Lambda, \beta^{\vee}\rangle >1$ for any simple root $\beta$, then $\pi$ is non-unitary.