Winter School 2016 in Representation Theory of Reductive Groups
2016年1月22日(金)-27日(水)に 東京大学大学院数理科学研究科（駒場キャンパス）で開催します.主要講師は3人
- Jeffrey Adams (University of Maryland)
- Yiannis Sakellaridis (Rutgers University and the National Technical University of Athens)
- Tobias Weich (Paderborn University)
- Wan-Yu Tsai (Institute of Mathematics Academia Sinica)
プログラムの暫定版
| Jan. 22 (Fri.) Room 126 | |
| 10:00-11:00 | Yiannis Sakellaridis |
| 11:20-12:20 | Tobias Weich |
| 14:00-15:00 | Yiannis Sakellaridis |
| （15:30-16:30 TBA） | |
| Jan. 23 (Sat.) Room 126 | |
| 10:00-11:00 | Yiannis Sakellaridis |
| 11:20-12:20 | Tobias Weich |
| 14:00-15:00 | Yiannis Sakellaridis |
| （15:30-16:30 TBA） | |
| Jan. 25 (Mon.) Room 126 | |
| 10:00-11:00 | Jeffrey Adams |
| 11:20-12:20 | Yiannis Sakellaridis |
| 14:00-16:00 | Jeffrey Adams |
| 16:30-17:30 | Tobias Weich |
| Jan. 26 (Tue.) Room 128 | |
| 10:00-11:00 | Jeffrey Adams |
| 11:20-12:20 | Yiannis Sakellaridis |
| 14:00-16:00 | Jeffrey Adams |
| 16:30-17:30 | Wan-Yu Tsai |
| Jan. 27 (Wed.) Room 126 | |
| 10:00-11:00 | Jeffrey Adams |
| 11:20-12:20 | Yiannis Sakellaridis |
| 14:00-16:00 | Jeffrey Adams |
| （16:30-17:30 TBA） |
タイトルとアブストラクト
- Yiannis Sakellaridis (Rutgers University and the National Technical University of Athens)
Mini-course on the Relative Langlands ProgramAbstract: In this series of talks, aimed primarily at graduate students who have some familiarity with reductive groups and harmonic analysis, I will outline a generalization of the Langlands program inspired from the work of Jacquet, D. Prasad and others. The basic objects here are spherical varieties, a very interesting class of varieties with a group action that generalizes symmetric spaces and has been studied systematically by Brion, Knop, Luna, Vust and others. Familiarity with the Langlands program will not be required: it will be obtained as a special case, when the spherical variety is a reductive group H, thought of as a symmetric space under the action of H x H by left and right multiplication.
Schedule of lectures:
- 1) Introduction to automorphic forms in the language of adeles.
- 2) Basic structure theory of spherical varieties.
- 3) Representation theory and harmonic analysis on reductive groups and spherical varieties.
- 4) Global period integrals and their conjectural factorization.
- 5) Generalizations of the Casselman-Shalika formula, and the L-function of a spherical variety.
- 6) The relative trace formula.
- 7) Local and global harmonic analysis on algebraic stacks.
- Tobias Weich (Paderborn University)
Wavefront sets of unitary Lie group representationAbstract: Wavefront sets (or the singular spectrum) are a central tool in microlocal analysis and they have been introduced around 1970 by Sato and H{\"o}rmander in the context of PDEs in order to measure singularities of distributions. Around 1980 Kashiwara-Vergne and Howe showed how one can transfer the notion of wavefront sets from distributions to unitary representations of Lie-groups and that this notion is very useful to study questions in harmonic analysis.
In this lecture series we will first introduce the notion of wavefront sets for distributions and illustrate their meaning by a few examples. We will then introduce the notion of the wavefront sets of a unitary representations and its use in representation theory. In particular we will emphasize the relation to the orbit method. Finally we will focus on a recent joint work with Benjamin Harris and we will see how techniques for the regularization of oscillating integrals can be used to determine the wavefront sets of induced representations.
- Wan-Yu Tsai (Institute of Mathematics Academia Sinica)
Some small genuine representations of a nonlinear cover and the model orbitAbstract: Let G be the real points of a simply laced, simply connected complex Lie group, and G~ be the nonlinear two-fold cover of G. We will discuss a set of small genuine representations of G~, denoted by Lift(C), which can be obtained from the trivial representation of G by a lifting operator. The representations in Lift(C) can be characterized by the following properties: (a) the infinitesimal character is $\rho/2$; (b) they have maximal tau-invariant; (c) they have a particular associated variety \O. We will use the split real group of type $D_{2n}$ as an example to exhibit the following further consequences: (1) all representations in Lift(C) are parametrized by pairs (central character, real form of \O); (2) The K-structure of the regular functions on a real form of \O is multiplicity free and matches up with the K-types of the small representations in Lift(C) attached to that real orbit.
- Jeffrey Adams (University of Maryland)
Atlas of Lie Groups and RepresentationsAbstract: The Atlas of Lie Groups and Representations is a software project to compute the unitary dual of an arbitrary real Lie group. I will be giving on overview of the project in a series of 6 lectures (9 hours total). I will give some mathematical background, and spend some time on a tutorial on getting started using the software. I will use the software during the presentation. It is not necessary, but users can download and install the software at www.liegroups.org.