#### 2018 Automorphic forms, representations of Lie groups

#### 2018 Automorphic forms, representations of Lie groups

#### Content

**July 5-6. Introductory lectures**

**July 5, 2018, 9:00-12:00. N202**

Chengbo Zhu: Representation theory of real reductive groups:

algebraic aspect.

Abstract: I will explain

(1) The basic theory of (g;K)-modules;

(2) A construction of the fundamental series;

(3) Unitarisable modules with nonzero (g;K)-cohomology;

(4) Invariants of Harish-Chandra modules.

References:

(1) Real Reductive Groups, I, by Nolan Wallach

(2) Associated varieties and unipotent representations, by David

Vogan.

**14:30 - 17:30. N202**

Binyong Sun: Representation theory of real reductive groups:

analytic aspect

Abstract: I will explain some basic notions and basic facts in

representation theory of real reductive groups, emphasizing on

the analytic aspect. Basic knowledges on topological spaces, smooth

manifolds and Lie groups are required. I hope to cover the

following materials:

1. Topological vector spaces and function spaces;

2. Representations and smooth representations;

3. Casselman-Wallach representations and smooth automorphic

forms;

4. Generalized matrix coefficients and characters;

5. Discrete series representations, tempered representations and

Langlands classification;

6. Generalized Whittaker models.

References:

N. R. Wallach: Real Reductive Groups, I, Academic Press Pure

and Applied Mathematics, Boston, 132 (1988).

N. R.Wallach: Real Reductive Groups II, Academic Press, Boston,

1992.

**July 6, 9:00-12:00. N202**

Kai-Wen Lan: An example-based introduction to Shimura varieties

Abstract: I will start by explaining why Shimura varieties are

natural generalizations of the more classical objects called modular

curves, and give many examples with explicit descriptions of

groups and symmetric domains. If time permits, I will also introduce

their boundary components, and discuss about their models

over rational numbers or even integers. The lectures will be for

people who are not already familiar with these topics—for most of

them, some willingness to see matrices larger than 2x2 ones should

suffice. (I hope to allow simple factors of all possible types A, B,

C, D, and E to show up if time permits, but it is not necessary to

know beforehand what this means.)

References: Introductory lecture notes,

http://www.math.umn.edu/ kwlan/articles/intro-sh-ex.pdf

**14:30 - 17:30. N202**

Genkai Zhang: Complex analysis and representation theory related

to Hermitian symmetric spaces.

Abstract: Hermtian symmetric spaces and their quotient spaces

form an important class of complex manifolds and they appear

in geometry as moduli spaces and in number theory. I’ll start

with their geometric/algebraic definitions and explain the following

topics:

1. Harish-Chandra realization, Siegel domain realization, and related

realization in projective spaces,

2. Bergman spaces, Bergman metric and kernels, the Hua-Kostant-

Schmid decomposition and reproducing kernel expansion.

3. Introduction of general holomorphic discrete series generalizing

Bergman spaces.

4. Plancherel formula on compact and non-compact Hermitian

symmetric spaces.

References:

1. S. Helgason, Differential geometry, Lie groups and Symmetric

spaces.

2. J. Faraut and A. Koranyi, Analysis on symmetric cones.

**July 9-13. Workshop**

**July 9, 9:30-10:30, N202**

Jing-Song Huang: Dirac operators, orbit method and unipotent

representations.

Abstract: The method of coadjoint orbits for real reductive

groups is divided into three steps in cooperation with the Jordan

decomposition of a coadjoint orbit into hyperbolic part, elliptic

part and nilpotent part. This is formulated in Vogan’s 1986

ICM plenary speech. The hyperbolic step and elliptic step are

well understood, while the nilpotent step to construct unipotent

representations in correspondence with 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.

**11:00-12:00, N202**

Chufeng Nien: Converse theorem on distinction.

Abstract: Twisted Gamma factors of a (generic or cuspidal) representation

against all generic representations encode information

about the representation . Converse theorem reveals how

many twisted Gamma factors we should consider in order to single

out a unique isomorphism class. This talk is about the relation

between special values of twisted Gamma factors and GLn(F)-

distinguished representations of GLn(E), where E is a quadratic

extension of F.

**14:30-15:30, N219**

Zhi Qi: Bessel identities over the complex field.

Abstract: A fundamental class of special functions in the number

theory and representation theory for GL2(R) is classical Bessel

functions (over R+). Certain (integral) formulae for Bessel functions

have deep representation theoretic interpretations. For example

the formulae of Weber and Hardy on the Fourier transform

of Bessel functions on R may be considered as a realization of

the Waldspurger correspondence over R. In this talk, we show

how certain Bessel identities in the Waldspurger correspondence

over C follow from the Weber-Hardy type formula over C.This is

a joint work with Jingsong Chai.

**15:45-16:45, N219**

Lei Zhang: The exterior cubic L-function of GU(6) and unitary

automorphic induction

Abstract: In this talk, we will discuss an extension of Ginzburg-

Rallis’ integral representation for the exterior cube automorphic

L-function of GL(6) to the quasi-split unitary similitude group

GU(6).

Furthermore, we introduce the automorphic induction for GU(n)

and show that those exterior cube L-functions have poles if and

only if the cuspidal representations are automorphically induced

from GU(3).

**July 10, 9:15-10:15, N202**

Kai Wang: An introduction to holomorphic isometries on symmetric

domains.

Abstract: In this talk, we will survey some recent progress on

holomorphic isometries on symmetric domains. We will construct

isometric holomorphic embeddings of the unit ball into higher rank

symmetric domains in an explicit way using Jordan triple systems,

and prove uniqueness results for domains of rank 2, including the

exceptional domain of dimension 16.

**10:30-11:30, N202**

Pavle Pandzic: Dirac index and twisted characters

Abstract: Dirac operators have played an important role in representation

theory of real reductive Lie groups since the work of

Parthasarathy and Atiyah-Schmid on the construction of discrete

series representations in the 1970s.

One of the important invariants of representations is the Dirac

index. An algebraic way to define the Dirac index is as the Euler

characteristic of the Dirac cohomology of the associated Harish-

Chandra module. The concept of Dirac cohomology was introduced

by Vogan and subsequently studied by Huang-Pandzic and

others. One of the important properties of the Dirac index of a

representation in the equal rank case is its close relationship with

the character on the compact Cartan subgroup.

In the unequal rank case, the Dirac index of all representations

is zero and therefore it is a useless notion. We have however

introduced a new invariant, twisted Dirac index, which is a good

substitute for the classical notion in the unequal rank cases. In this

lecture I will first review some basic facts about representations,

Harish-Chandra modules, Dirac cohomology and index. I will

then explain the notion of twisted Dirac index and present some

examples and applications. This is joint work with Dan Barbasch

and Peter Trapa.

**14:30-15:30, N202**

Chengbo Zhu: Orbit method and unipotent representations

Abstract: A fundamental problem in representation theory is to

determine the unitary dual of a given Lie group G, namely the set

of equivalent classes of irreducible unitary representations of G.

A principal idea, originated in a groundbreaking paper of A. A.

Kirillov in the sixties, is that there is a close connection between

irreducible unitary representations of G and the orbits of G on

the dual of its Lie algebra. This is known as the orbit method.

In this talk, I will describe basic ideas of the orbit method as well

as a recent development on the problem of unipotent representations,

which is to associate unitary representations to nilpotent

coadjoint orbits and which is the hardest part of the orbit method.

We solve this problem for real classical groups, by combining analytic

ideas of R. Howe on theta lifting and algebro-geometric

ideas of D. A. Vogan, Jr. on associate varieties. Geometrically,

our construction is guided by Kraft-Procesi construction of closures

of nilpotent conjugacy classes in classical Lie algebras. This

is joint work with J.-J. Ma and B. Sun.

**15:45-16:45, N202**

Dihua Jiang: Automorphic Bessel Descents and Related Problems

Abstract: The theory of endoscopic classification of the discrete

automorphic spectrum of classical groups provides a fundamental

structure for automorphic representations. The theory of automrphic

descents is to understand the refined structure and properties

of automorphic representations, based on the endoscopy theory.

In this talk, we will discuss the progress on theory of automorphic

Bessel descents from my joint work with Lei Zhang and also with

Baiying Liu and Bin Xu.

**July 11, 9:15-10:15, N202**

Jun Yu: Geometric interpretation of the Kirillov conjecture

Abstract: The Kirillov conjecture in the 1960s assers that the

restriction of any irreducible unitary representation of GL(n,K)

(K=R or C) to a microbolic subgroup is irreducible, which is

shown by Sahi (for tempered reppresentations) and Baruch (in

general) after 40 years. In this talk we give a geometric interpreation

of this conjecture in the framework of Kirillov-Duflo’s

orbit method. This is joint work with Gang Liu (France).

**10:30-11:30, N202**

Sidhartha Sahi: The Capelli eigenvalue problem

**Afternoon: Free discussions.**

**July 12, 9:15-10:15, N202**

Birgit Speh: Restrictions of representations of rank one orthogonal

groups : results and applications.

Abstract: I will discuss the restriction of infinite dimensional representations

of the rank one orthogonal group O(n+1,1 ) to O(n,1)

and discuss the space of symmetry breaking operators for any pair

of irreducible representations of G and the subgroup G0 with trivial

infinitesimal character. I will discuss the application of this

result to the multiplicity conjecture by B.Gross and D.Prasad for

tempered principal series representations of (SO(n+1,1), SO(n,1))

with trivial infinitesimal character .These results also allow us to

find periods of irreducible representations of the Lorentz group

with nonzero-cohomologies. This is joint work with T. Kobayashi.

**10:30-11:30, N202**

Dongwen Liu: Tower property and theta lifting for loop groups

Abstract: We study the theta lifting for loop groups and extend

the classical tower property established by S. Rallis to the loop

setting. As an application we give the first examples of nonvanishing

cusp forms on loop groups. This talk is based on a joint

work with Yongchang Zhu.

**14:30-15:30, N202**

E. Sayag: Counting of lattice points and Harmonic Analysis

Abstract: The classic lattice counting problem in the plane

(Gauss) and in the hyperbolic plane (Delsarte) insures a close

relationship between the number of lattice points in a Ball of radius

R and the volume of that Ball. Similar problems were studied

in the 90s for symmetric spaces using Harmonic analysis (Duke-

Rudnick-Sarnak) and Ergodic methods (Eskin-Mcmullen). Since

then many researchers extended these results to cover other homogeneous

spaces and studied the error term of the counting.

We shall give an overview of the Harmonic analysis method and

then focus on obtaining effective bounds on the error term in the

counting in the case of reductive spherical spaces. Our results are

based on a novel comparison of Lp norms of Eigenfunctions on

these spaces.

Joint work with Bernhard Kroetz and Henrik Schlichtkrull (Acta

Mathematica Sinica, 2018).

**15:45-16:45, N202**

Zhuohui Zhang: Functions on U(2) and the (g;K)-Module

Structure of Principal Series

Abstract: I will describe the differential action of g on the principal

series representations of G = SU(2; 1) and Sp(4;R). The

structure of the principal series for G can be described explicitly

by the g-action on the matrix coefficients of the maximal compact

subgroup K G. The computation, based on the Clebsch-

Gordan coefficients of the representations of K, is algorithmic and

generalizable to the real reductive groups with a maximal compact

subgroup isogenous to a product of multiple copies of SU(2) and

U(1). As an application of the machinery, I will describe some

subquotients of the principal series, their restrictions to K, and I

will also use the combinatorial properties of hypergeometric functions

to calculate the intertwining operators of minimal principal

series.

**July 13, 9:00-10:00, N202**

Kai-Wen Lan: Local systems over Shimura varieties: a comparison

between two constructions.

Abstract: Given a Shimura variety X associated with some algebraic

group G, and some algebraic representation V of G (satisfying

some conditions when restricted to the center), we can define

two kinds of filtered vector bundles with integrable connections,

over X. The first one is based on the classical complex analytic

construction using double quotients, while the second one is a new

p-adic analytic construction based on the p-adic Riemann-Hilbert

correspondence in the recent work by Ruochuan Liu and Xinwen

Zhu. We know how to relate these two when the local systems are

given by the relative cohomology of some family of abelian varieties

over X. But what should we do when X is a general Shimura

variety, in which case no convenient family of algebraic varieties

(or, rather, "motives") are available? A priori, the complex and

p-adic analytic constructions have very little in common. In this

talk, we shall review the background materials and formulate the

problem more precisely, and give an answer. (This is joint work

with Hansheng Diao, Ruochuan Liu, and Xinwen Zhu.)

**10:30-11:30, N202**

Binyong Sun: Cohomologically induced distinguished representations

Abstract: Cohomological parabolic induction, which is closely

related to complex analysis, provides a general method to construct

irreducible representations of real reductive groups. We

give a general construction of local periods on these cohomologically

induced representations.

**Afternoon: Free discussions.**