17 problems

Guichard-Labourie-Wienhard conjecture. The set of $\Theta$-positive representations $\rho:\pi_1(\Sigma_g)\to G$ is open and closed in $\operatorname{Hom}(\pi_1(S),G)/G$. In particu…

Let $G_{q\widetilde{q}}^+(\mathbb{R})$ be the positive quantum group associated with $\mathfrak{g}_\mathbb{R}$, and let $L^2(G_{q\widetilde{q}}^+(\mathbb{R}))$ carry the left and r…

Cayley-component conjecture. Cayley components are all the connected components of the space of $\Theta$-positive representations.

Let $p>2$ and consider the smooth connected components of the space $\textnormal{Pos}_{p,p+1}(\Gamma)$ of $\Theta$-positive representations of a surface group $\Gamma$ into…

Classification conjecture. The class of all irreducible regular positive representations of $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ is classified by these five families.

Transcendental-relations conjecture. The transcendental relations are satisfied for all Chevalley generators of $\mathcal{U}_q(\mathfrak{sl}(n+1,\mathbb{R}))$ with $\lambda$ chosen…

Let $\Sigma$ be a closed oriented surface and let $(\mathrm{G},\mathrm{P}^\mathbb{R}_\Theta)$ admit a $\Theta$-positive structure. A representation…

Let $\mathcal{P}_\lambda^J$ be an arbitrary parabolic positive representation of $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$, and let its Casimir operators have spectrum in the ambien…

Let $\mathcal{P}_\lambda^J$ be a parabolic positive representation, let $\mathcal{X}_q^{\mathbf{D}(\overline{\mathbf{i}})}$ be its associated doubled quantum torus, and let…

Let $\mathfrak{D}_q(\mathfrak{g})$ be the quantum group embedded in the quantum torus algebra $\mathcal{X}_q^{\mathbf{D}(\mathbf{i})}$, and let a cluster mean any quantum torus alg…

McShane equality conjecture. The non-strict inequality

Let $G$ be any Lie group, and let $P$ be a complex probability defined on $G$. Suppose that $P$ is spanned by a set of well-behaved representations of $G$, for example, bounded rep…

Let $S$ be a closed orientable surface, let $\mathrm{G}$ be a semisimple Lie group, and let $\mathrm{P}^{\Theta}\subset\mathrm{G}$ be a parabolic subgroup such that…

Let $\mathcal{U}_{q\widetilde{q}}(\mathfrak{b}_{\mathbb{R}})$ be the split real quantum Borel subalgebra, and let $\mathcal{P}=\mathcal{P}_{\vec{0}}$ denote the positive representa…

Let $\mathcal{U}_{q\widetilde{q}}(\mathfrak{g}_\mathbb{R})$ be the modular-double quantum group and let $\mathcal{P}_{\alpha}$ and $\mathcal{P}_{\beta}$ be positive representations…

Characterization conjecture. The positivity and transcendental relations completely characterize the class of positive representations for $U_{q\tilde{q}}(\mathfrak{g}_\mathbb{R})$…

Let $\textbf{U}_{\mathfrak{q}\til[\mathfrak{q}]}(\mathfrak{g}_\mathbb{R})$ be the modular double for a split real quantum group, and consider its positive principal series represen…