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.