Workshop on Quantization and Representation Theory
Aug 3-7, 2018, Sun Yat-sen University
Purpose:
This workshop will be about representation theory of complex/real reductive Lie groups and related topics, with special focus on coadjoint orbit method, realization of unipotent representations and their relationship with deformation quantization and algebraic geometry. We are aiming to bring together researchers to communicate on recent progress.
Organizers:
HU, Jianxun (Sun Yat-sen University)
LEUNG, Naichung Conan (The Chinese University of Hong Kong)
LI, Changzheng (Sun Yat-sen University)
YU, Shilin (Taxes A&M University)
Speakers:
ADAMS, Jeffrey (University of Maryland)
BARBASCH, Dan (Cornell University)
FU, Baohua (Chinese Academy of Sciences)
HUANG, Jing-Song (Hong Kong University of Science and Technology)
LEUNG, Naichung Conan (The Chinese University of Hong Kong)
LI, Qin (South University of Science and Technology of China)
SUN, Binyong (Chinese Academy of Sciences)
VOGAN, David (Massachusetts Institute of Technology)
WONG, Kayue Daniel (Cornell University)
YU, Shilin (Taxes A&M University)
Program:
Venue: Room 415 (New Math. Building)
Aug 2: Arrival day.
Aug 8: Departure day.
Remark: every 90-minute slot consists of a 60-minute talk and 30-minute informal discussions.
| Aug 3 (Fri) | Aug 4 (Sat) | Aug 5 (Sun) | Aug 6 (Mon) | Aug 7 (Tue) | |
| 9:30-10:30 | |||||
| Vogan1 | Leung/Yu | Vogan2(start from 9:00) | Free discussions | Free discussions | |
| Tea Break | |||||
| 10:40-11:40 | Question session (Vogan) | Question session (Leung/Yu) | Li(end up to 12:10) | ||
| Lunch Break | |||||
| 14:00-15:30 | Huang | Barbasch1 | Fu | Free discussions | (15:00-16:30) Barbasch2 |
| Tea Break | |||||
| 15:40-17:10 | Sun | Adams1 | Adams2 | (16:40-18:10) Wong | |
Banquet: 18:00-20:00 (Aug 4 )
Title and Abstracts:
Title1: Atlas of Lie Groups and Representations(by J. Adams)
Abstract: The goal of the Atlas project is to compute the Unitary Dual of a Lie group, and more generally to make computations in Lie theory accessible. I will give an overview of the present state of the project, and an indication of ongoing projects.
Title 2: Special Unipotent Representations(by J. Adams)
Abstract: Suppose G(R) is a real form of a connected complex reductive group G, and G^v is the complex dual group. qAccording to conjectures of Jim Arthur, associated to a complex nilpotent orbit of G^v is an "Arthur packet" of "unipotent" representations of G(R). I will give a definition of these packets, and sketch an algorithm to compute them.
Title: TBA (by D. Barbasch)
Abstract: TBA.
Title: On Fano complete intersections in rational homogeneous varieties(by B. Fu)
Abstract: Complete intersections inside rational homogeneous varieties provide interesting examples of Fano manifolds. We first classify these Fano complete intersections which are locally rigid. It turns out that most of them are hyperplane sections. We then classify general hyperplane sections which are quasi-homogeneous. This is a joint work with Chenyu Bai and Laurent Manivel.
Title: Dirac operators, orbit method and unipotent representations (by J.-S. Huang)
Abstract: The method of coadjoint orbits for real reductive groups is divided into three steps in correspondence to the Jordan decomposition of a linear functional into hyperbolic part, elliptic part and nilpotent part. Based on the theory established by Kirillov, Kostant and Duflo, Vogan formulated a program of the orbit method for reductive Lie groups in his 1986 ICM plenary speech. The hyperbolic step and elliptic step are well understood, while the nilpotent step to construct unipotent representations from nilpotent orbits has been extensively studied in several different perspectives over the last thirty years. Still, the final definition of unipotent representations remains to be mysterious. The aim of this talk is to show that our recent work (joint with Pandzic and Vogan) on classifying unitary representations by their Dirac cohomology shed light on what kind of irreducible unitary representations should be defined as unipotent.
Title: A new perspective on orbit method (by N.C. Leung/S. Yu)
Abstract: The coadjoint orbit method of Kirillov and Kostant suggests that irreducible unitary representations of a Lie group G should arise as geometric quantization of coadjoint orbits of G. The work of Barbasch, Vogan and many others suggest that one can attach `unipotent representations’ to nilpotent orbits of reductive Lie groups, which however are beyond the reach of geometric quantization. In this talk, we propose a new construction of unipotent representations using deformation quantization.
Title: Peak sections, deformation quantization and its representations (by Q. Li)
Abstract: In this talk, I will describe how deformation quantization of a Kahler manifold can be obtained from its peak sections. In particular, we will also obtain Hilbert space representations of deformation quantization algebras.
Title: Distinguished unipotent representations (by B. Sun)
Abstract: Let G be a real orthogonal group or a real symplectic group. We construct all special unipotent representations of G attached to distinguished nilpotent orbits of the Langlands dual group of $G$, by using theta lifts. We also show the unitarity of these representations.
Title1: Orbit method and unitary representations (by D. Vogan)
Abstract: The Kirillov-Kostant "philosophy of coadjoint orbits" says that for a Lie group G, there should be a close connection between the set G^ of irreducible unitary representations of G, and the set of "coadjoint orbits" of G on the dual of its Lie algebra. Unitary representations are interesting and useful because they are the building blocks of harmonic analysis: of understanding the action of G on spaces of functions (in the same way that the Fourier transform helps to understand functions on the real line). But unitary representations are difficult to find. Coadjoint orbits are much more elementary and familiar. If G is GL(n,R), the group of n x n invertible real matrices, then a coadjoint orbit is just a conjugacy class of n x n real matrices. Such conjugacy classes can be very explicitly and completely described by linear algebra. I'll explain a bit of the motivation (from mathematical physics) for the philosophy of coadjoint orbits, and something about what is known towards implementing it.
Title2: Orbit method for nilpotent orbits (by D. Vogan)
Abstract: Suppose G is a real reductive group. Work of Mackey, Harish-Chandra, Zuckerman, and others provides methods to attach unitary representations to semisimple coadjoint orbits of G on the dual g^* of its Lie algebra. To complete the orbit method, what is missing is a way to attach unitary representations to nilpotent coadjoint orbits. I will talk about recent work that at least makes it possible to tell when you have succeeded: given a unitary representation, to find the nilpotent coadjoint orbits to which it might be attached.
Title: Admissible Modules and Normality of Classical Nilpotent Varieties (by K. D. Wong)
Abstract: In the early 2000s, Ranee Brylinski construsted (g,K)-modules with the property that their K-structure matches the structure of regular functions on classical nilpotent varieties. In this talk, we give a description on the composition factors of these (g,K)-modules. In particular, we decompose the ring of regular functions of these nilpotent varieties into irreducible, algebraic modules. This verifies the conditions of normality of such varieties given by Kraft and Procesi. This is a joint work with Dan Barbasch.
Accommodations and Local information:
Hotels on campus:
SYSU hotel & Conference Center（中山大学学人馆）
Address: North gate of Sun Yat-sen University, Binjiang Dong Road, Haizhu District, Guangzhou
Tel: 020-89222888 Website: http://www.syskaifeng.com
Contact information:
Ms. YU, Jinyun
Email: yujin250 AT 163.com
Prof. LI, Changzheng
Email: lichangzh AT mail.sysu.edu.cn
Phone: +86-20-84111738
School of Mathematics, Sun Yat-sen University, No. 135, Xingang Xi Road, Haizhu, Guangzhou, 510275, China