## Lie Groups and Representation Theory Seminar 2019

List of speakers:
Takaaki Nomura, Yves Benoist #1, Clemens Weiske, Yves Benoist #2, Quentin Labriet,
 集中講義 intensive lectures (901-57 数理科学特別講義Ⅲ/0505115 数理科学続論C (理学部) 共通講義) Date: June 17 (Mon)-21 (Fri), 2019 Place: Room 123, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Takaaki Nomura 野村隆昭 (九州大学名誉教授/大阪市立大学数学研究所特別研究員) Title: 球面調和函数と群の表現 Abstract: 球面調和函数は，数学をはじめ，物理学や工学等，様々な分野で現れる重要な函数である．$\mathbb{R}^3$における2次元球面の場合は，ルジャンドル多項式を用いて，3次元回転群の表現論とともに詳しく論じられ，扱っている著書や文献も数多くある． 本講義では，次元を一般にした場合の球面調和函数について，古典的理論から群の表現論や非可換調和解析等の現代的視点までを一貫した形で概説する予定である． 集中講義 intensive lectures (数理科学特論VIII) Date: Oct 3, 10, 17, 24, 31, Nov 7, 14, 21, 28, 2019, 13:00-14:45 Place: Room 128, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Yves Benoist (CNRS, Université Paris-Sud) Title: Kleinian groups Abstract: Kleinian groups are discrete groups of isometries of the hyperbolic space. We will study both subgroups of finite and infinite covolume. Here are a few aspects that we will discuss: - The hyperbolic space, its boundary and its isometries. - Construction of discrete subgroups and lattices. - Limit set, convex cocompact subgroups and autosimilar fractal sets. - Hausdorff dimension, critical exponent and spectrum of the Laplacian. Date: Oct 23 (Wed), 2019, 16:30-18:00 Place: Room 128, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Clemens Weiske (Aarhus University) Title: Symmetry breaking and unitary branching laws for finite-multiplicity pairs of rank one Abstract: [ pdf ] Let $(G,G’)$ be a real reductive finite multiplicity pair of rank one, i.e. a rank one real reductive group $G$ with reductive subgroup $G’$, such that the space of symmetry breaking operators (SBOs) between all (smooth admissible) irreducible representations is finite dimensional. We give a classification of SBOs between spherical principal series representations of $G$ and $G’$, essentially generalizing the results on $(O(1,n+1),O(1,n))$ of Kobayashi-Speh (2015). Moreover we show how to decompose unitary representations occurring in (not necessarily) spherical principal series representations of $G$ in terms of unitary $G’$ representations, by making use of the knowledge gathered in the classification of the SBOs and the structure of the open $P’$ orbit in $G/P$ as a homogenous $G’$-space, where $P’$ is a minimal parabolic in $G’$ and $P$ is a minimal parabolic in $G$. This includes the construction of discrete spectra in the restriction of complementary series representations and unitarizable composition factors. 談話会 Colloquium Date: Oct 25 (Fri), 2019, 15:30-16:30 Place: Room 123, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Yves Benoist (CNRS, Université Paris-Sud) Title: Arithmeticity of discrete subgroups Abstract: By a theorem of Borel and Harish-Chandra, an arithmetic group in a semisimple Lie group is a lattice. Conversely, by a celebrated theorem of Margulis, in a higher rank semisimple Lie group $G$ any irreducible lattice is an arithmetic group. The aim of this lecture is to survey an arithmeticity criterium for discrete subgroups which are not assumed to be lattices. This criterium, obtained with Miquel, generalizes works of Selberg and Hee Oh and solves a conjecture of Margulis. It says: a discrete irreducible Zariski-dense subgroup of $G$ that intersects cocompactly at least one horospherical subgroup of $G$ is an arithmetic group. Date: Oct 30 (Wed), 2019, 16:30-18:00 Place: Room 128, Graduate School of Mathematical Sciences, the University of Tokyo Speaker: Quentin Labriet (Reims University) Title: On holographic transform Abstract: [ pdf ] In representation theory, decomposing the restriction of a given representation $\pi$ of a Lie group $G$ to an appropriate subgroup $G'$ is an important issue referred to as a branching law. In this context, one can define symmetry breaking operators, as $G'$-intertwining operators between the restriction $\pi|_{G'}$ and its irreducible components. Going in the opposite direction gives rise to holographic operators and the notion of holographic transform. I will illustrate this construction by two examples : - the diagonal case where one considers the restriction problem for $\pi$ being an outer product of two holomorphic discrete series representations, $G=SL(2,R)\times SL(2,R)$ and $G'=SL(2,R)$. - the conformal case for the restriction of a scalar valued holomorphic discrete series representation $\pi$ of $G=SO(2,n)$ to $G'=SO(2,n-1)$. I will then explain different methods for an explicit construction of such holographic operators in these cases, and present some of my results and open problems in this direction.