104 problems

A minimal simple $W$-algebra is one of the unitary minimal simple $W$-algebras under consideration, and an extremal (massless) module is an untwisted highest weight module of the e…

Let $k$ be $\mathbb{R}$ or $\mathbb{C}$, and let $P_n(k)$ be the mirabolic subgroup of $\mathrm{GL}_n(k)$. For an irreducible unitary representation $\pi$ of $\mathrm{GL}_n(k)$, co…

Let $G_{\mathbb{R}}$ be a real reductive group, and consider the two classes of unipotent $G_{\mathbb{R}}$-representations defined in the source, including birationally rigid unipo…

A complex reductive group has special unipotent representations, namely the representations introduced by Adams, Barbasch, and Vogan following ideas of Arthur. Unitarity conjecture…

Let $T$ be a regular tree and let $G\leqslant\operatorname{Aut}(T)$ be unimodular and transitive on the vertices of $T$. The group C$^*$-algebra of $G$ is CCR if it is liminal, a p…

Let $G_{d53d}$ be a connected reductive complex Lie group, let $G$ be a real form of $G_{d53d}$, let $\check G$ be its Langlands dual group, and let $\check{\mathfrak g}$ be the Li…

Vogan's unipotent representation conjecture. Unipotent representations are unitary and form the building blocks of the unitary dual.

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

Let $U(p,q)$ be the real reductive group of unitary matrices of signature $(p,q)$, and let $A_\mathfrak{q}(\lambda)$ denote a cohomologically induced module in the weakly fair rang…

Let $G$ be a locally compact group and let $u$ be a spread out probability measure on $G$. A $u$-boundary is a boundary associated with the measure $u$, and it carries an as…

Let $G_{\mathbb{R}}$ be a real reductive group, and let an irreducible Harish-Chandra representation of $G_{\mathbb{R}}$ with real infinitesimal character be realized as the global…

Let $G$ be a complex reductive Lie group, and let $X=G\cdot\lambda=G/G(\lambda)$ be a nilpotent coadjoint orbit. Suppose an irreducible $G$-equivariant local system on $X$ is given…

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 $G$ be a classical group of type I, and call a representation unitarily small in the sense defined in the paper. Salamanca-Riba–Vogan conjecture. Every irreducible unitary repr…

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…

Let $G$ be a real reductive group, and call an irreducible unitary representation $\pi\in\Pi_{\Lambda}(G)$ unitarily small if $\Lambda$ lies in the convex hull of…

Li's positivity conjecture. For unitary $\pi$, the form $(,)_{\pi}$ is always positive semidefinite.

The paper constructs irreducible unitary representations of a finitely generated free group that are weakly contained in the left regular representation and in which a specified li…

Ol'shanskii's conjecture. The other unitary groups considered in the paper should also be type I.

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

Let $\pi$ be an irreducible admissible representation of good parity. The good-parity Arthur-type conjecture. The representation $\pi$ is of Arthur type if and only if it is unitar…

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…

Mohammadi–Oh's conjecture. The optimal condition $\delta_\Gamma>d/2$ suffices for $L^2(\Gamma\backslash G)$ to admit a strong spectral gap for the complementary series, in arbitrar…