| Abstract: |
| The parabolic positive representations of $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ were previously constructed by quantizing the classical parabolic induction corresponding to arbitrary parabolic subgroups, such that the Chevalley generators act by positive self-adjoint operators on a Hilbert space. This generalizes the (standard) positive representations introduced earlier corresponding to the minimal parabolic (i.e. Borel) subgroup. In this talk, we will show how one can study the scalar actions of the generalized Casimir operators by certain reductions from the standard representations to the parabolic cases. |
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