Monday, April 4, 2022
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All day |
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4:00pm |
04/04/2022 - 4:00pm Rigidity phenomena in homogeneous spaces have been extensively studied over the past few decades with several striking results and applications. In this talk, we will give an overview of activities pertaining to the quantitative aspect of the analysis in this context. In particular, we will discuss recent density and equidistribution theorems in quotients of $SL_2(\mathbb C)$ and $SL_2(\mathbb R)\times SL_2(\mathbb R)$, which provide a polynomial error rate. This is based on joint works with Elon Lindenstrauss and Zhiren Wang. Location:
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04/04/2022 - 4:30pm Abstract: We construct a new family of irreducible representations of \mathcal{U}_q(\mathfrak{g}_\mathbb{R}) and its modular double by quantizing the classical parabolic induction corresponding to arbitrary parabolic subgroups, such that the generators of \mathcal{U}_q(\mathfrak{g}_\mathbb{R}) act by positive self-adjoint operators on a Hilbert space. This generalizes the well-established positive representations introduced by [Frenkel-Ip] which correspond to induction by the minimal parabolic (i.e. Borel) subgroup. We also study in detail the special case of type A_n acting on L^2(\mathbb{R}^n) with minimal functional dimension, and establish the properties of its central characters and universal \mathcal{R} operator. Finally we will explain a positive version of the evaluation module of the affine quantum group \mathcal{U}_q(\widehat{\mathfrak{sl}}_{n+1}) modeled over this minimal positive representation of type A_n. Ref: arXiv:2008.08589 Location:
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