10 problems

Tensor-product conjecture. The tensor product of regular positive representations is again regular, and it can be decomposed into a direct integral of regular positive representati…

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…

Boundedness conjecture. The collection

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 $\mathcal{U}_{q\widetilde{q}}(\mathfrak{b}_{\mathbb{R}})$ 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 $U_{q\tilde{q}}(\mathfrak{g}_\mathbb{R})$…

Plancherel-type decomposition conjecture. The space $L^2(G_q)$ is decomposed into a direct integral of tensor products of the positive representations under the left and right regu…