# Workshop on Quantization and Representation Theory

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