Positive Representations of Split Real Quantum Groups and Future Perspectives

We construct a special principal series representation for the modular double $U_{{\mathfrak q} \bar {\mathfrak q}}(\mathfrak g\mathbb R)$ of type Ar representing the generators by positive essentially self-adjoint operators satisfying the transcendental relations that also relate q and $\tilde q$. We use the cluster variables parameterization of the positive unipotent matrices to derive the formulas in the classical case. Then we quantize them after applying the Mellin transform. Our construction is inspired by the previous results for $\mathfrak g_{\mathbb R}=\mathfrak {sl}(2,\mathbb R)$ and can be generalized to all other types of simple split real Lie algebra $\mathfrak g_{\mathbb R}$. We conjecture that our positive representations are closed under the tensor product and we discuss the future perspectives of the new representation theory following the parallel with the established developments of the finite-dimensional representation theory of quantum groups.