Stone–von Neumann conjecture for positive representations of the split real quantum Borel subalgebra

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 representation with zero parameter. A representation is required to satisfy the integrability relations

$$K_i^{\mathbf{i} b^{-1} s}\mathbf{e}_j^{\mathbf{i} b^{-1} t}=e^{-\pi a_{ij}st}\mathbf{e}_j^{\mathbf{i} b^{-1} t}K_i^{\mathbf{i} b^{-1} s}$$

and, in the simply-laced case,

$$\mathbf{e}_i^{\mathbf{i} b^{-1} s}\mathbf{e}_{ij}^{\mathbf{i} b^{-1} t}=e^{\pi st}\mathbf{e}_{ij}^{\mathbf{i} b^{-1} t}\mathbf{e}_i^{\mathbf{i} b^{-1} s},$$ $$\mathbf{e}_j^{\mathbf{i} b^{-1} s}\mathbf{e}_{ij}^{\mathbf{i} b^{-1} t}=e^{-\pi st}\mathbf{e}_{ij}^{\mathbf{i} b^{-1} t}\mathbf{e}_j^{\mathbf{i} b^{-1} s},$$

with analogous relations in the non-simply-laced case. Stone–von Neumann conjecture. Any irreducible representation of $\mathcal{U}_{q\widetilde{q}}(\mathfrak{b}_{\mathbb{R}})$ satisfying the conditions of a positive representation is unitarily equivalent to $\mathcal{P}$. This is proposed as a Stone–von Neumann theorem for the Borel subalgebra of a split real quantum group; the displayed relations encode the quantum-plane and Serre-type integrability conditions, while the corresponding classification remains open.