91 problems

Let $G/H$ be a homogeneous space in the reductive setting, and let $K$ be a maximal compact subgroup of $G$. Rank inequality conjecture. If $G/H$ admits a cocompact discontinuous g…

Saturation conjecture for semisimple Lie groups. If $\lambda+\mu+\nu$ annihilates all elements of $T$ with semisimple centralizer, then

Converse to Theorem 15. The converse of Theorem 15 also holds.

Singular-triple criterion. There exists a triple $\sigma\in {\mathcal P}(G)\cap L^3$ with $Tr(\sigma)\in Q(R)$ and $\sigma\notin Tens(G)$ if and only if there exists a singular tri…

The structure conjecture. (1) If $R$ is simply-laced, then

Let $H$ be a connected semi-simple Lie group without almost simple factors of type $\mathrm{A}_1$ and without compact factors. For sufficiently large $x$, consider maximal irreduci…

Matsuki–Gindikin conjecture. For every $K_\mathbb C$-orbit $S$ of nonholomorphic type on $X$, one has

Gray–Wolf conjecture. Every homogeneous strictly nearly Kähler space is a naturally reductive 3-symmetric space equipped with its canonical almost-complex structure.

Free-action quantization conjecture. A quasi-Poisson manifold $X$ for $({\mathfrak{g}},Z)$ can be quantized if the ${\mathfrak{g}}$-action on $X$ is free, i.e. if $X$ is a principa…

Let $A_n(s)$ be the alternating factorization defined in the paper, and let $\widetilde{\operatorname{SL}}(2,\mathbb{R})$ be the universal covering group of…

Let $n\in\mathbb{N}$ with $n\geq3$. Let $A_n(s)$ and $A(n;r,s,t)$ denote the alternating factorizations used in the paper, let $s_n$ be the associated switching parameters, and let…

Let $G$ be a Lie group, let $H$ be a closed subgroup, and let $d(H)$ denote the invariant used in the paper to measure the dimension relevant to tessellations of $G/H$. One-dimensi…

Nonexistence conjecture. None of these homogeneous spaces has a tessellation. These are presented as three special cases of Kobayashi's general conjecture; the supplied source give…

Let $G$ be a compact Lie group with Lie algebra $\mathfrak g$, let $G^c$ be its complexification, and let $\Omega$ be the data defining the examples associated to $G$. Denote the r…

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 $G$ be a connected Lie group, let $ath.E(G,\mathcal{D}(G))$ denote the conjugacy relation on the space $\mathcal{D}(G)$ of discrete subgroups of $G$, and call an equivalence re…

Let $G$) be an exponential solvable Lie group, and let ${\mathcal J}_1\subseteq{\mathcal J}_2$ be closed two-sided ideals of $C^*(G)$ such that ${\mathcal J}_2/{\mathcal J}_1$ is a…

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…

Schur critical-index conjecture. The critical index satisfies

Let $G$ be a simply connected Lie group, let $\tau_h$ be the involution associated with an Euler element $h\in\mathfrak g$, and let $(U,\mathcal H)$ be an antiunitary representatio…

Nonexistence conjecture. The homogeneous space

Let $\Gamma$ be a finite subgroup of $SU(2)$. For an Abelian central subgroup $Z\subset G$ of a connected compact simple Lie group $G$, set $H=G/Z$, with central extension … Let…

Let $\Gamma$ be a finite group, and let $G$ be a connected compact simple Lie group. Write $N(\Gamma,G)$ for the number of homomorphisms from $\Gamma$ to $G$ up to conjugation by…

The XZ conjecture for $G_2$. If

Let $N\geq 2$, and let $f:[0,1]^{\frac{N(N-1)}{2}}\times \mathbb{T}^{\frac{N(N+1)}{2}-1}\to\mathbb{C}$ be an admissible function, meaning that it has a finite expansion in the toru…