Workshop on Quantization and Representation Theory

发布人:周妍 发布日期:2018-07-18
活动时间
-
活动地址
新数学楼 415

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 (Chinese University of Hong Kong)
LI, Changzheng (Sun Yat-sen University)
YU, Shilin (Texas 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
(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
(Texas 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
Barbasch2
(15:00-16:30)
Tea Break Tea Break
15:40-17:10 Sun Adams1 Adams2
Wong
(16:40-18:10)
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 Gv is the
complex dual group. According to conjectures of Jim Arthur, associated to a complex nilpotent
orbit of Gv 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: Unipotent Representations and Unitarity, I and II (by D. Barbasch)
Abstract: Conjecturally, the unitary dual of a reductive group is obtained by unitarity preserving
constructions from "basic representations" on Levi components. This is consistent with the "orbit
method/quantization" which conjectures that unitary representations should be constructed from
data which are local systems on orbits in the coadjoint representations. The unitarity preserving
constructions are unitary induction, derived functors, and complementary series. The first two
correspond to the semisimple part of the Jordan decomposition of an element in the dual of the Lie
algebra. The first indication of what the basic representations would be, come from conjectures of
Arthur about the residual spectrum of locally symmetric spaces. They are called "special unipotent
representations", and are given in terms of the dual group. The description of the unitary dual of
complex classical groups necessitates enlarging this class to "unipotent representations". They are
parametrized by data in the dual group, as well as by data related to coadjoint orbits. In the first talk
I will describe these parametrizations for complex classical groups and their covers. A special focus
will be on real versions of these parametrizations, notably Sp(p,q) and SO(2n)*. In the second talk I
will describe some results on the special case of nonlinear orthogonal groups. A particular focus
will be ways of counting admissible representations with special properties that characterize
unipotent representations. As time permits, I will discuss other cases of real groups.
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 × 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 constructed (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:
Hotel 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
Workshop 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