REPRESENTATION THEORY OF LIE GROUPS
(July 2002 – January 2003)
~ Abstracts ~
Character theory of nonlinear
real groups
Jeffrey Adams, University of Maryland, College Park, USA
This is joint work with Peter Trapa and Rebecca Herb,
and related to the talks of David Renard. I will discuss
two closely related phenomena: Vogan duality and character
lifting (along the lines of Shelstad's theory) for
nonlinear real groups. The case of the metaplectic group
Mp(2n,R) and the twofold cover of GL(n) are related to the
thetacorrespondence and KazhdanPatterson lifting
respectively. A general theory, at least in the case of one
root length, is available.
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unitarity@home
Jeffrey Adams, University of Maryland, College Park, USA
The unitary dual of a real Lie group is only known in
some special cases. For any given Lie group there is a
finite algorithm to compute the unitary dual. I will
discuss a proposal to compute the unitary dual of real Lie
group by computer.
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Representations of padic Groups
Gordan Savin, University of Utah, USA
This set of tutorials provide an elementary
introduction, with exercises, to representations of padic
reductive groups.
We start with definitions and preliminary results on padic
fields, structure of GL_n(F) over a padic field F, and
smooth representations. In order to keep the exposition as
simple as possible, we restrict ourselves to GL_2(F).
However, the topics and their proofs are chosen so that
they easily generalize to GL_n(F) and other reductive
groups.
Next, we introduce induced and cuspidal representations,
and prove that irreducible smooth representations are
admissible. We shall also discuss the composition factors
of induced representations, unitarizable representations,
and construct the complementary series for GL_2(F).
Finally, we go back to GL_n(F) and describe the
composition factors of (regular) induced representations.
This result, due to Rodier, is based on the combinatorics
of the root system, and the reduction to the special case
of GL_2(F). As such, it gives a good introduction to
further, more advanced topics.
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Dirac operators in
representation theory
Jing Song Huang, Hong Kong University of Science and
Technology, Hong Kong and Pavle Pandzic, University of
Zagreb, Croatia
Dirac operators are widely used in the index theory in
differential geometry and geometric construction of
discrete series representations. The aim of this tutorial
is to reveal the algebraic nature of Dirac operators,
namely Vogan's conjecture on Dirac cohomology. We will
explain a proof of this conjecture and show its wide
applications in representation theory. We will also include
some background material on Lie groups representations and
Lie algebra cohomology.
The 10 hour lectures are tentatively arranged as
follows:
 (g,K)modules
 Clifford algebra & Spinors
 Dirac operators & group representations
 Aq(lambda)modules
 Vogan's Conjecture and its proof
 BorelWeil Theorem and Discrete Series
 Lie algebra cohomology
 (g,K)cohomology
 Dirac cohomology and other cohomologies
 Multiplicity of automorphic forms
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Orbits of Lie Groups
HungYean Loke, National University of Singapore,
Singapore
We will cover the basic definitions and classifications
of semisimple and nilpotent orbits. If time permits, we
will describe in detail some interesting exceptional orbits
and their coordinate rings.
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Degenerate Principal Series
Representations of GL(n,C)
SooTeck Lee, National University of Singapore, Singapore
Let G=GL(n,C) and P the parabolic subgroup of G with its
Levi factor isomorphic to GL(nk,C)xGL(k,C). We consider
the representation of G induced from a character of P. We
shall calculate explicitly the action of the Lie algebra of
G on this representation. This allows us to determine its
reduciblitly, composition series and unitarity.
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Uncertainty and Entropy in
TimeFrequency: Finite versus Continuous
Tomasz Przebinda, University of Oklahoma, Norman, USA
In 1957 Hirschman proved that the sum of entropies of a
function f (with
) and its Fourier
transform is nonnegative, [Hi]. He also observed that a
stronger version of this inequality:
(proven later by Beckner [B, page 177]) implies the
HeisenbergWeyl Uncertainty Principle. Hirschman
conjectured that the minimizers for the sharp inequality
(1) were Gaussians, as is the case for the HeisenbergWeyl
Uncertainty Principle. We shall show that this is indeed
the case.
There is an analog of (1) for functions f
defined on a finite abelian group, with applications in
Signal Processing. The shall also describe these minimizers.
They depend on the structure of the finite abelian group
and are not "Gaussians" or discretized Gaussians.
This discrepancy between the finite and the continuous case
seems to be unexpected in the Signal Processing community.
In fact the main motivation for this work is to expose this
fundamental difference.
We shall spend considerable amount of time explaining
the notions involved, including the entropy and the
representation theory of the groups we shall need.
References:
[B] W. Beckner, Inequalities in Fourier Analysis,
Annals of Mathematics 102 Number 6 (1975), 159  182.
[Hi] I. I. Hirschman, Jr., A Note on Entropy, Amer.
J. Math. 79 (1957), 152  156.
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An explicit trace formula
Benedict Gross, Harvard University, USA
In this talk, we will illustrate some recent progress
that has been made on the trace formula, by Arthur and
Kottwitz. We will give a formula for the Euler
characteristic of the cohomology of the discrete spectrum.
This integer is given as the sum of rational numbers,
indexed by stable torsion conjugacy classes. The main
contribution of each class is a product of certain values
of Artin Lfunctions, at negative integers.
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A comparison of Haar measures
Benedict Gross, Harvard University, USA
A locally compact topological group G has a
leftinvariant measure dµ, unique up to scalar multiples.
For G compact, we can normalize the measure so that
.
For G discrete, we can normalize so that µ(g)
= 1, for all g in G.
In certain situations, which we will consider, there are
two naturally defined invariant measures on G, and we will
want to determine their ratio. For example, if G is finite,
it is both compact and discrete. The ratio of the measures
given above is just the order of G. We will consider the
situation when G is the group of points of a reductive
algebraic group over a finite or local field, and when G is
the group of adelic points of a reductive algebraic group
over a global field. The ratios are computed using
Lfunctions associated to G.
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Mappings which preserve
familes of curves
Michael Cowling, University of New South Wales,
Australia
This talk is a survey of some results in geometry from
Darboux to Tits. First, any bijection of the plane which
maps lines to lines is affine, i.e., a composition of a
linear map and a translation. There are many local versions
of this result: on one hand, a bijection of the open unit
square which preserves line segments is linear, while on
the other, there are nonlinear (but fractional linear) maps
of the open unit circle preserving chords.
Next, we consider the plane and the sphere. All
bijections of the projective plane which preserve
projective lines come from the projective group, while
there are many maps of the sphere which preserve great
circles which are not even continuous.
Finally, we consider maps of threedimensional space
which preserve two families of lines, one family of lines
parallel to the y axis and the other family of lines lying
in planes parallel to the xz plane and with gradients equal
to the y coordinates. It is shown that these maps are
affine, and that this implies (for SL(3,R)) a theorem of
Tits that the morphisms of a spherical building come from
the group. A local version of this result is also outlined.
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The Geometry of K/M
Michael Cowling, University of New South Wales,
Australia
The analytic/geometric construction of complementary
series realizes these exceptional unitary representations
on spaces of Sobolev type. Indeed, in the classical case of
SO(1,n), the kernels of the intertwining operators may be
expressed in terms of the euclidean distance and the
corresponding representations may be represented on Sobolev
spaces defined using the classical laplacian. In the
general rank one case, an analytic description of these
spaces relies on a noneuclidean distance and a sublaplacian.
In the rank one case, these fit into the notyetmature
theory of CarnotCaratheodory manifolds.
In the higher rank case, it is less clear what is the
appropriate geometry for K/M, or more generally G/P where P
is a parabolic subgroup. This talk, based on joint work
with Filippo De Mari, Adam Koranyi and Hans Martin Reimann,
is about this problem. What is the notion of conformal
geometry appropriate to the higher rank case? We give
elementary proofs of some results of Yamaguchi on this
question.
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Homomorphisms of the
icosahedral group into reductive groups
George Lusztig, MIT, USA
In the representation theory theory of finite groups one
studies homomorphisms from a finite group into GL_n up to
conjugacy. More generally GL_n could be replaced by any
reductive group. We are mainly interested in the case where
the finite group is the icosahedral group. The most
interesting case is that where the reductive group is of
type E_8. In this case we find the complete classification
of homomorphisms up to conjugacy completing earlier work of
D.D.Frey.
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Ktypes and character expansions
Julee Kim, University of Michigan, Ann Arbor, USA
Let k be a padic field, and let G be a group of
kpoints of a connected reductive group defined over k. For
an irreducible admissible representation of G, we discuss
an asymptotic expansion of its character. This expansion
depends on Ktypes contained in the given representation.
This is a joint work with Fiona Murnaghan.
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Dual blobs and Plancherel
formulas
Julee Kim, University of Michigan, Ann Arbor, USA
Continuing from the previous talk, we discuss some
applications of character expansions.
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Theta lifting of nilpotent
orbits for symmetric pairs
ChenBo Zhu, National University of Singapore, Singapore
We consider a reductive dual pair (G, G') in the stable
range with G' the smaller member. We study theta lifting of
nilpotent K'_C orbits, where K' is a maximal compact
subgroup of G, and describe the K_C module structure of the
regular function ring of the closure of the lifted
nilpotent orbit of the symmetric pair (G, K). Connection
with the Ktype structure of theta lift of unitary
representations of G' will also be discussed. This is joint
work with Kyo Nishiyama of Kyoto University.
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Signatures of invariant
Hermitian forms: general theory
David Vogan, MIT, USA
Let G be a real reductive Lie group, K a maximal compact
subgroup, and X an irreducible HarishChandra module for G.
By a theorem of HarishChandra, X comes from a unitary
representation of G if and only if X admits an invariant
Hermitian form that is positive definite.
By a theorem of Knapp, it is easy to tell when X admits
an invariant Hermitian form. I will explain how to define a
"signature" for such an invariant form, and what
such signatures can look like in general. One can then
formulate the problem of computing the signature of any
invariant Hermitian form. This problem includes the problem
of determining the unitary dual of G. I'll explain a few
results and many open problems about this computation.
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Computing signatures of
invariant Hermitian forms
David Vogan, MIT, USA
Let G be a reductive group over a local field k, and P =
LN a minimal parabolic subgroup of G. Suppose K is a
compact subgroup of G with the property that G = KP and the
intersection of K with P is a subgroup M of L. (This allows
us to identify the homogeneous spaces G/P and K/M.) Write
a^* for the (real vector space) of characters of L taking
positive real values; such characters are trivial on M.
Given a character nu in a^*, we extend nu to a character
of P by making the unipotent radical N act trivially, and
then form the principal series representation I(nu) = Ind_P^G
(nu). These representations are defined on a common Hilbert
space H = L^2(K/M).
Except in case nu=0, the representation I(nu) is not
unitary with respect to the natural Hilbert space structure
on H. However, one can sometimes find a nice family
<,>_nu of hermitian forms on H, with the property
that I(nu) preserves the form <,>_nu. If the form
<,>_nu is semidefinite, one finds in this way a
unitary composition factors of I(nu); these are
"spherical complementary series" representations.
One would therefore like to know exactly when the forms
<,>_nu are semidefinite. When G is split and k is a
nonarchimedean field, Barbasch and Moy have shown that the
semidefiniteness of <,>_nu can be made into a
question in the group algebra of the Weyl group. One
beautiful and immediate consequence is that the answer does
not depend on the field k.
I will explain the result of Barbasch and Moy. Part of
their condition is that a certain matrix depending on nu
(of size equal to the dimension of a^*) be positive
semidefinite. The rank of this matrix was computed by
Joseph some fifteen years ago; now one wants to know
whether its nonzero eigenvalues are all positive.
Finally, I will discuss work with Salamanca and Barbasch
providing consequences of these calculations for the
archimedean fields R and C.
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Harmonic analysis for padic
groups
Stephen Debacker, University of Chicago, USA
This set of tutorials will attempt to provide an
elementary introduction to harmonic analysis on padic
groups. This area of mathematics is very technically
demanding. Thus, in order to keep the primary focus on the
ideas rather than the details, we will restrict our
attention to the general linear group GL_n, and, when
required for expository reasons, we will further restrict
our attention to the group GL_2. Everything we discuss is
valid in a far more general context.
We start by introducing the basic objects: orbital
integrals, characters of irreducible admissible
representations, and Fourier transforms (on the Lie algebra
of our group). The ultimate goal is to understand the
HarishChandraHowe local character expansion which
provides a very beautiful connection among these objects.
We next restrict our attention to nilpotent orbital
integrals. In this context, we discuss a fundamental result
of Deligne and RangaRao which states that orbital
integrals define invariant distributions, and we will then
turn our attention to Huntsinger's proof that the Fourier
transform of an orbital integral is represented by a
locally integrable function on the regular set.
Next we discuss a version of Howe's conjecture for the
Lie algebra. Note: This "conjecture" has been a
theorem for well over 25 years.
Finally, we pull everything together by discussing the
HarishChandraHowe local character expansion. This is a
very deep result which states that on some neighborhood of
the identity, the character of an irreducible addmissible
representation can be written as a linear combination of
the Fourier transforms of nilpotent orbital integrals.
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Branching laws for padic
groups
Dipendra Prasad, Mehta Research Institute, India
In these lectures we will introduce the concept of
sphercal pairs, i.e., pair of groups (G,H) with G reductive
such that the action of H on the flag variety of G/B has an
open orbit. In this situation, finite dimensional
representations of the algebraic group G have at most 1
dimensional space of Hinvariant vectors. We will look at
some examples, and see what happens for infinite
dimensional representations of padic groups. We will try
to give some examples, and some proofs.
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Invariant differential operators
on a
classical lie supergroup
Tomasz Przebinda, University of Oklahoma, Norman, USA
An invariant eigendistribution on a real
reductive group is a distribution which is in variant
under the conjugation by elements of the group, and is an
eigendistribution for the commutative algebra of left and
right invariant differential operators on the group. A real
reductive dual pair, together with the underlying
symplectic space, may be viewed as a classical Lie
supergroup. We shall show how does the notion of an
invariant eigendistribution extend to this context. In
particular we shall recover the correspondence of
infinitesimal characters for representations under Howe's
correspondence.
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Invariant eigendistributions on
a classical lie supergroup
Tomasz Przebinda, University of Oklahoma, Norman, USA
The goal of this lecture is to explain the
HarishChandra's method of descent in the context of a
classical Lie supergroup and to apply it to study the
characters of representa tions which occur in Howe's
correspondence. We shall also explain a conjectural
relation between these characters, which may be traced back
to the Cauchy determinant identity.
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Degenerate Principal Series
Representations of U(p,q)
SooTeck Lee, National University of Singapore, Singapore
Download/view PDF
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Generic representations of
reductive groups over finite rings
George Lusztig, MIT, USA
Abstract. Let G be a connected reductive algebraic group
defined over a finite field F_q. Let r be a strictly
positive integer. We are concerned with the study of
complex representations of the finite group G(F_q[[Z]]/(Z^r))
where Z is an indeterminate. Using a cohomological method (etale
cohomology) extending that of Deligne and the author
(1976), we construct an irreducible representation of
G(F_q[[Z]]/(Z^r)) for any "maximal torus" and a
generic character of it; for r at least 2, this was stated
without proof in a paper I wrote in 1977.
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Group schemes associated to
MoyPrasad groups
JiuKang Yu, University of Maryland, USA
We construct canonical smooth integral models of padic
reductive groups associated to groups in the theory of
MoyPrasad and SchneiderStuhler, and study their
properties.
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What are minimal
representations of padic groups?
Gordan Savin, University of Utah, USA
In this talk we shall compare several definitions of
minimal representations over padic fields. Next, we shall
describe an approach to minimal representations due to
Weissman, and discuss how its properties can be used to
establish results on dual pair correspondences.
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Commuting differential
operators with regular singularities at infinity and
completely integrable quantum systems
Toshio Oshima, University of Tokyo, Japan
In this talk we discuss commuting differential operators
defined at an infinite point of a Cartan subgroup. They
include HeckmanOpdam hypergeometric systems, differential
equations satisfied by Whittaker vectors, finite Toda
chains etc. We give their classification and their explicit
expression.
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Cycle Spaces and Representations
Joseph Wolf, University of California at Berkeley, USA
I'll recall some geometric constructions of tempered
representations of real reductive Lie groups, and some
possibilities for going to nontempered representations.
The latter will concentrate on double fibration transform
methods, e.g. the complex Penrose transform, and the role
of the linear cycle space of a flag domain.
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Structure of the Cycle Spaces
Joseph Wolf, University of California at Berkeley, USA
I'll describe some very recent results that give the
precise structure of the linear cycle space of a flag
domain. Roughly speaking, all the famous tubular
neighborhoods of G/K, in its complexification, coincide;
and they are equal to the cycle space except in certain
specific cases. The implications for double fibration
transforms will be discussed.
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Supercuspidal character germs
of classical and other groups
Jeffrey Adler, University of Akron, USA
Supercuspidal character germs ought to be expressible as
an explicit linear combination of Fourier transforms of
elliptic orbital integrals. In most cases, only one orbit
should be involved. We will show how to prove this result
for many supercuspidal characters of many groups.
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Base change, with an example
Jeffrey Adler, University of Akron, USA
Given some knowledge of characters and of BruhatTits
theory, it is possible to make base change more explicit in
certain cases. We look at an example involving base change
from U(3) to GL(3). This is joint work in progress with
Joshua Lansky.
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On selfdual representations of
padic groups
Dipendra Prasad, Mehta Research Institute, India
An irreducible selfdual representation of a group
carries an invariant bilinear form which is unique up to
scaling. It is therefore either symmetric or
skewsymmetric. It is a classical theorem for compact Lie
groups that the bilinear form is symmetric or
skewsymmetric depending on the action of an element in the
centre of the group of order less than or equal to 2. In
this talk we look at the analogue of this theorem for padic
groups. We summarise the results for generic
representations of quasisplit groups which appeared in
IMRN (1999) which gives a rather close analogue for many
groups, including groups with trivial centre, and classical
groups such as GL(n), Sp(n), SO(n), but works only for SL(n)
when n is not congruent to 2 modulo 4. There are
counterexamples for such an expectation for SL(6).
We formulate a conjecture for general quasisplit group
in terms of lifting of the corresponding Langlands
parameter to a certain 2 fold cover of the Lgroup.
As a specific nonquasisplit group, we discuss the case
for division algebras, and state the conj. made with
Dinakar Ramakrishnan which relates the parity of the
bilinear form of an irreducible selfdual representation of
the invertible elements of a division algebra in terms of
the corresponding information on the Galois representation.
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Tensor Product of Degenerate
Series and Theta Correspondence
EngChye Tan, National University of Singapore, Singapore
We shall describe an intertwining map from the
oscillator representation to the tensor product of two
degenerate principal series representations of G and G'
(which form a reductive dual pair in the sense of Howe). We
shall give examples of results on local theta
correspondence for some pairs.
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Equidistribution of Cycles on
Hilbert Modular Varieties
JianShu Li, Hong Kong University of Science and
Technology, Hong Kong
The AndreOort conjecture is an assertion on the Zariski
density of CMpoints on Shimura varieties. This can be
viewed as an equal distribution property of zero
dimensional Shimura subvarieties. In this talk we shall
discuss equal distribution of subvarieties on Hilbert
modular varieties defined by quaternion algebras. This is a
report of joint work in progress with D. Jiang and S.
Zhang.
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A Vanishing Result for the
Cohomology of Arithmetic Groups
JianShu Li, Hong Kong University of Science and
Technology, Hong Kong
I will outline a proof of the following vanishing
theorem (joint work with Schwermer): Let $G$ be a
semisimple Lie group and $K$ a maximal compact subgroup.
Let $$ q_0(G) = \frac{1}{2}[\dim G/K  rank(G) + rank(K)]
$$ Suppose $E$ is a finite dimensional representation of
$G$ with regular infinitesimal character and $\Gamma$ is an
arithmetic subgroup of $G$. Then $H^j(\Gamma, E)=0 $ for
all $j<q_0(G)$.
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Homogeneity Results for
Reductive padic Groups
Stephen Debacker, University of Chicago, USA
In the 1970s Roger Howe made two "finiteness"
conjectures concerning invariant distributions on reductive
padic groups and their Lie algebras. These conjectures
were answered in the affirmative for the Lie algebra by
HarishChandra and Howe and for the group by Clozel (in
characteristic zero) and Barbasch and Moy (in general). In
the 1990s Waldspurger proved a very precise version of the
Howe conjecture for the Lie algebra for classical
unramified groups. In this talk, we will discuss a
generalization of Waldspurger's result and discuss possible
applications.
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Quantum Analogues of
Coherent Families at Roots of 1
R. Parthasarathy, School of Mathematics, Tata Institute
of Fundamental Research
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Theta lifting, affine
quotients, and degree formula for highest weight modules I.
Kyo Nishiyama, Kyoto University, Japan
We begin with a compact dual pair, and give the
Bernstein degree of unitary highest weight modules using a
theorem of invariant theory. After that, we will show how
to generalize the idea to noncompact dual pair. We explain
how to lift nilpotent orbits (or associated variety), and
realize them as an affine quotient of a bigger nilpotent
orbit which is lifted from the trivial orbit. This talk is
based on a joint work with Chengbo Zhu (NUS).
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Theta lifting, affine
quotients, and degree formula for highest weight modules
II.
Kyo Nishiyama, Kyoto University, Japan
Let (G, G') be a dual pair of type I. In the stable
range, we give a formula of lifting of nilpotent orbits.
There are many interesting examples of lifted nilpotent
orbits, and we explain some application to branching rules
and multiplicity free actions. Finally, we give an exact
formula of the associated cycle of the theta lifting of
unitary highest weight modules in the stable range. This
talk is based on a joint work with Chengbo Zhu (NUS).
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Dual pair PGL(3) x G_2 in E_6,
and (g_2, SL(3))modules
Gordan Savin, University of Utah, USA
Let G_2 be the exceptional complex Lie group, and g_2
its Lie algebra. Since SL(3) is a spherical subgroup of
G_2, the theory of (g_2, SL(3))modules is a good one. In
this lecture we describe a classification of irreducible
(g_2, SL(3))modules, and construct a correspondence
between (algebraic) representations of complex PGL(3) and
(g_2, SL(3))modules using a "minimal
representation" of E_6. On the level of infinitesimal
characters the correspondence is functorial for the
inclusion of dual groups SL(3) > G_2. This then give
us Capellitype identities in the eveloping algebra of E_6,
modulo the Joseph ideal, such as C + 2=3C' + 14, where C
and C' are the Casimir operators of PGL(3) and G_2,
respectively.
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On embeddings of derived
functor modules into degenerate principal series I & II
Hisayosi Matumoto, University of Tokyo, Japan
The definition of the Sato hyperfunction suggests
existence of embeddings of derived functor modules into
degenerate principal seires as boundary value maps.
However, it seems difficult to construct an embedding of a
global cohomolgy such as a derived functor module via a
local method. However, in some case, we can show existence
(or nonexistence) of such a hopedfor embedding of a
derived functor module via another method. In this talk, I
would like to explain, in particular, the complex group
case. In this case, existence of a homomorphism between
generalized Verma modules is seriously related to existence
of a embeddings of a defived fuctor module.
In part II, I will discuss examples of embedding of
derived functor modules into degenerate principal series.
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Isotropy representations
for singular unitary highest weight modules
Hiroshi Yamashita, Hokkaido University, Japan
We describe the isotropy representation ${\mathcal
W}_\lambda$ attached to every singular unitary highest
weight module $L(\lambda)$. In the oscillator setting, it
has been already shown that the assignment ${\mathcal
W}_\lambda^\ast \leftrightarrow L(\lambda)$ essentially
gives the Howe duality correspondence with respect to a
compact dual pair. In this talk, We focus our attention on
$L(\lambda)$'s which can not be realized by the theta
correspondence. By using the projection onto the PRVcomponent,
the isotropy representations are explicitly determined for
such highest weight modules. This gives in particular a
clear understanding of the multiplicity formulae obtained
by Kato and Ochiai for the cases BI, DI and EVII. Moreover,
it turns out that the representation ${\mathcal W}_\lambda$
is irreducible for every singular unitary highest weight
module. This is a joint work with Akihito Wachi of Hokkaido
Institute of Technology.
Download/view PDF
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Eigenspace representations of
symmetric spaces of exceptional type
Hiroyuki Ochiai, Tokyo Institute of Technology, Japan
For a classical, complex, or rankone symmetric space,
every invariant differential operator comes from the center
of the corresponding universal enveloping algebra. The
classification of the symmetric pairs which do not have
this property is completed by Helgason in 90s with some
additional information. In this talk, I'll discuss a more
detailed structure of the image of the center of the
enveloping algebra inside the ring of invariant
differential operators and the structure of the eigenspace
representations. I'll also mention the relation with the
work by HuangOshimaWallach on the generalized
eigenfunctions on such symmetric spaces.
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Theta correspondence,
onedimensional representations, and unitarity
Annegret Paul, Western Michigan University, USA
I will discuss the theta correspondence of unitary
onedimensional representations, starting with the case of
the dual pairs (U(p,q),U(r,s)) (this is joint work with
Peter Trapa). In particular, I'll talk about the questions
of occurrence, unitarity and Langlands parameters of the
theta lifts, and which families of unitary (unipotent?)
representations occur.
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On Finite Subgroups of Some
Linear Groups
Sun Binyong, Hong Kong University of Science and
Technology
The so called torsion conjecture says: Given any abelian
variety A defined over a number field k, the order of the
torsion part of A(k) is bounded by a constant C(k, d) which
only depends on the number field k and the dimension d of
the abelian variety. If we replace the abelian variety by
certain linear groups, the analogue is much easier. We get
some results on this direction.
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Local Langlands conjecture for
nonlinear real Lie groups
Peter E. Trapa, University of Utah, USA
Let G~ be a nonlinear double cover of a linear reductive
real Lie group G. The purpose of this talk is to establish
some connections between the representation theory of G and
representation theory of an associated linear real group H.
(H turns out to be a real form of a subgroup of the
Langlands dual of the complexification of G.) This includes
a matching of parameters for representations of G~ and H
which is compatible, in an appropriate sense, with
stability. For instance, when G = Sp(2n,R), we find that H
= O(p,q). In this case, the matching of parameters is
closely related to the theta correspondence, and the
appropriate stabilization recovers a lifting of Adams. As a
further example, when G = GL(n,R), we obtain a version of
KazhdanPatterson lifting. This is joint work with Jeffrey
Adams and David Renard. This talk is an overview; later
talks of Renard will provide significantly more details and
results.
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New restrictions on
characteristic cycles of HarishChandra modules
Peter E. Trapa, University of Utah, USA
We discuss various structures  some geometric, some
algebraic  on the computation of characteristic cycles of
HarishChandra modules. This leads to many new
computations. For instance, we find families of
representations whose characteristic cycles have (roughly)
2^{r/2} irreducible components, where r is the split rank.
The calculations suggest a kind of duality for the leading
piece of the characteristic cycle.
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Ktype Structure of Degenerate
Principal Series
Roger Howe, Yale University, USA
This talk will show in several examples, how the
analysis by Howe and Lee, of the most degenerate principles
series for GL_n (R) and GL_n (C), can be used to find
interesting information about more general principal
series. This is in part joint work with ST Lee, and part a
report on calculations by C. Will.
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Small unitary representations
Siddhartha Sahi, Rutgers University, USA
We describe explicit models of small unitary
representations of certain semisimple Lie groups and
establish a thetatype correspondence arising from their
tensor products.
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Deformation quantization and
invariant distributions
Siddhartha Sahi, Rutgers University, USA
The exponential map carries invariant germs of
distributions on the Lie algebra to those on the Lie group.
An conjecture of KashiwaraVergne asserts that after a
certain universal twist (motivated by the orbit method)
this is actually an isomorphism for the (partial) algebra
structures on the two spaces. We describe a proof of this
result based on Kontsevich's starproduct construction.
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Representations with Scalar
Ktypes and Applications
ChenBo Zhu, National University of Singapore, Singapore
We discuss some results of Shimura on invariant
differential operators and extend a folklore theorem about
spherical representations to representations with scalar
Ktypes. We then apply the result to obtain nontrivial
isomorphisms of certain representations arising from local
theta correspondence, many of which are unipotent in the
sense of Vogan.
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Conformal geometry and
analysis on minimal representations of O(p,q)
Toshiyuki Kobayashi, RIMS, Kyoto, Japan
We apply methods from conformal geometry of pseudoRiemannian
manifolds to a general construction of an infinite
dimensional representation of the conformal group on the
solution space of the Yamabe equation. Then, I will discuss
various geometric models together with explicit inner
products of the minimal representation of O(p,q).
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Parameterizations via BruhatTits
Theory
Stephen Debacker, University of Chicago, USA
We shall discuss various parameterizations which arise
naturally from BruhatTits theory.
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Cohomology of Compact
Quotients of Hermitian Domains and the Role of the Modules
$ A_{q,}$
R. Parthasarathy, School of Mathematics, Tata Institute
of Fundamental Research
Some irreducible unitary representations contribute to
the cohomology of compact quotients of symmetric spaces via
'Matsushima's formula'. We recall them and describe some
known facts about them. Towards the end, we will briefly
discuss some results of T.N. Venkataramana on restriction
of cohomology classes to some submanifolds arising from
reductive subgroups and secondly some results about the
Hodge decomposition (for the hermitian case).
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Bernstein center distributions
Marko Tadic University of Zagreb, Croatia
Bernstein center of a reductive $p$adic group is an
analogue of the center of the enveloping algebra of a Lie
algebra. It has a number of very important applications.
The center is usually studied as regular functions on the
algebraic variety of (infinitely many) connected
components. It is isomorphic to the space of invariant
distributions which have essentially compact support (this
means that these distributions are compactly supported
after convolution with any locally constant compactly
supported function). The isomorphism is given by
"Fourier transform".
The delta distribution supported at identity is in
Bernstein center. It is not so obvious to point out some
other elements and a question is how to describe more
explicitly this huge family of distributions. This is topic
of our talk. The results that we shall present are obtained
jointly with A. Moy (they are part of a longer joint
project).
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Macdonald Positivity
Conjecture, n! Conjecture, GHilbert Scheme and Diagonal
Harmonics
Claudio Procesi, Universita' di Roma, Italy
In 89 Macdonald discovered a remarkable class of
symmetric polynomials depending on 2 parameters and
conjectured that, once expanded in terms of Schur
polynomials, they have coefficients which are polynomials
in the two parameters with positive integer coefficients.
Garsia and Haiman developed a strategy to prove this
conjecture using a certain bigraded representation of the
symmetric group, finally this led to the n! conjecture
which has been recentely solved by Haiman using deep
properties of the Hilbert scheme. The same techiques have
allowed to solve conjectures on diagonal harmonics.
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The associated variety of a
unipotent representation
Dan Barbasch, Cornell University, USA
The associated cycle of a HarishChandra module is a
linear combination with integer coefficients of nilpotent
orbits. The associated variety is the (closure of the)
union of the orbits which occur with nonzero multiplicity.
One use of the associated cycle is via results of AdamsBarbaschVogan
which show how to construct stable (in the sense of
Langlands) combinations of characters. Another use is to
derive information about composition series of induced
representations.
In this talk I will describe the cycle of the special
unipotent representations in the case of the classical real
algebras sp(n) and so(p,q). As a consequence one obtains a
description of the associated variety of each
HarishChandra cell.
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The geometry of admissible
representations
Jing Song Huang, Hong Kong University of Science and
Technology, Hong Kong
In this talk we would like to show the connection
between the moment map for the associated variety and the
condition being admissible for group representations.
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Branching of Minimal Holomorphic Representations
Zhang Genkai, University of Chalmers, Sweden
We study the tensor product decomposition of a minimal
holomorphic representation with its complext conjugate. It
has continuous part and in certain cases also a descrete
parts consisting of complementary series. We find some
(new) unitary spherical representations. We compute the
ClebschGordan coefficients and study the quantization of
the complementary series representations.
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The Structure of the Ring of
Quasisymmetric Polynomials
Nolan Wallach, University of California at San Diego,
USA
The ring of quasisymmetric polynomials nvariables is a
subring, QS, of the polynomials that contains the symmetric
polynomials, S. In this lecture I will explain the
ingredients of a proof (due to the speaker and A.Garsia) of
a conjecture of Bergeron and Reutenauser that says that QS
is free as an Smodule. This implies, in particular, that
QS is a CohenMacaulay ring. The proof involves a very
interesting element of the group algebra of the symmetric
group which has played an important role in the speaker’s
work on Jacquet integrals and in Diaconis’s shuffling
theory.
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Branching Coefficients of
Holomorphic Representations
Zhang Genkai, University of Chalmers, Sweden
We study the restriction to real forms $H/L$ of bounded
symmetric domains $G/K$ of the (analytic continuation of)
scalar holomorphic discrete series. We find $L$invariant
holomorphic polynomials in terms of the Jack symmetric
polynomials and we compute their SegalBargmann and
spherical transforms.
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Branching Rules and Covariants
of Qubits
Roger Howe, Yale University, USA
The current interest in quantum computation provides a
new set of questions in invariant theory. This talk will
provide a brief introduction to the ideas of quantum
computation, including a desription of the qubit, the basic
unit of quantum information. It will then describe some
computations in invariant theory which may help elucidate
the structure of small systems of qubits.
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KazhdanLusztig Algorithm for
Nonlinear Real Reductive Groups and Applications to
Functoriality
David Renard, University of Poitiers, France
Nonlinear groups appear in Automorphic Forms theory,
for instance via Howe's dual pairs correspondences or
KazhdanPatterson lifting. Thus, it is natural to try to
extend Langlands formalism to these groups, and to study
the "functoriality" of the above correspondences.
Over the real numbers, we will use the geometric
reformulation of Langlands local conjectures due to AdamsBarbaschVogan
to achieve this goal for double covers of reductive linear
groups (e.g. the metaplectic group). This consists of three
steps:
 Establish KazhdanLusztig algorithm.
 Establish Vogan's character multiplicity duality.
 Give applications to functoriality on various
examples.
Although there is no dual group or $L$group in the
picture, the existence of Vogan's duality for nonlinear
groups is sufficient for our purposes. It allows us to
define $L$packets and to state functoriality principles.
This is joint work with P. Trapa.
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Holomorphic Continuation of
Generalized Jacquet Integrals
Nolan Wallach, University of California at San Diego,
USA
In this lecture some generalizations of the speaker’s
earlier work on holomorphic continuation of Jacquet
integrals that are general enough to apply to the work that
he did on quaternionic representations. In this
generalization one must replace standard Bruhat theory with
an extension due to Kolk and Varadarajan.
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Measures of entanglement in
quantum computing
Nolan Wallach, University of California at San Diego,
USA
This lecture will give a rapid introduction to quantum
computing for mathematicians. The emphasis will be on the
notion of entanglement and measures of entanglement. Some
new applications of invariant theory will be discussed.
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Classification of Some Classes
of Irreducible Representations of Classical padic Groups
Marko Tadic, University of Zagreb, Croatia
The set of tutorials provide an introduction to
classification of some important series of irreducible
representations of general linear and classical groups
(having in mind unitary representations). We shall deal
more with padic groups, but we shall also discuss real
groups (some of the results are uniform). One of the goals
will be to give an introduction to classification modulo
cuspidal data of irreducible square integrable
representations of classical padic groups. Further, we
shall talk about unitary duals of general linear groups
(and corresponding proofs). We will finish with some
questions regarding unitary representations of classical
groups.
The topics which we plan to cover are following:
 Harmonic analysis and unitary duals
 Nondiscrete locally compact fields, classical groups,
reductive groups
 Kfinite vectors
 Smooth representations
 Parabolically induced representations
 Jacquet modules
 Filtrations of Jacquet modules
 Square integrable and tempered representations
 Langlands classification
 Geometric lemma and algebraic structures
 Square integrable representations of general linear
groups
 Two simple examples of square integrable
representations of classical groups
 Invariants of square integrable representations of
classical groups
 Reduction to cuspidal lines
 Parameters of representations supported in cuspidal
lines
 Integral case
 Nonintegral case
 On local Langlands correspondences
 Unitary duals of general linear groups
 On unitarizability problem for classical groups
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Equivariant Analogues of
Zuckerman and Bernstein Functors
Pavle Pandzic, University of Zagreb, Croatia
There are two well known constructions of HarishChandra
modules: Zuckerman's derived functor construction, and
BeilinsonBernstein localization theory. They both use some
homological algebra, however in different categories which
makes the relations between them nonobvious. Both
constructions can however be performed in the framework of
equivariant derived categories introduced by
BeilinsonGinzburg.
In the first talk I will review the notion of (ordinary)
derived category, and define the equivariant derived
category of HarishChandra modules. In the second talk I
will explain how to get analogs of Zuckerman and Bernstein
functors in the equivariant derived category setting. These
functors are related by a version of "Hard
Duality" of Knapp and Vogan, which I will sketch at
the end.
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Dirac Operators, Lie Algebra
Cohomology and Group Representations
Jing Song Huang, Hong Kong University of Science and
Technology, Hong Kong
The aim of this talk is to show the connection between
Dirac cohomology and Lie algebra cohomology of unitary
representations.
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The Classical Voronoi
Summation Formula
Wilfried Schmid, Harvard University, USA
Abstract: The Voronoi summation formula provides
explicit formulas for sums of the type $\sum_n f(n) a_n$,
where $a_n$ is an arithmetically defined sequence of
coefficients and $f(x)$ a compactly supported function of
bounded variation. It has become an essential tool in
analytic number theory. Originally established on a
casebycase basis,it is now regarded as a statement about
Lfunctions satisfying a functional equation, and is
deduced from the functional equation. I shall discuss the
formula, its proof and some applications.
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A Local Property of
Distributions, with Applications to Lfunctions
Wilfried Schmid, Harvard University, USA
I shall describe a local property of distributions 
"vanishing to infinite order" at a point  which can be
used to establish functional equations in various contexts.
Automorphic distributions for SL(2,R), i.e., boundary
distributions of modular forms and Maass forms, have this
property, as do certain onevariable distributions derived
from automorphic distributions on higher rank groups. This
is joint work with Steve Miller.
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Lfunctions and Voronoi
summation for GL(3)
Wilfried Schmid, Harvard University, USA
Certain Lfunctions attached to automorphic forms for
GL(3,Z), which are important from the point of view of
analytic number theory, do not satisfy a functional
equation. The classical approach to Voronoi summation does
not work in this case. I shall state a Voronoi summation
formula for these Lfunctions, sketch its proof, and
describe an application. The same type of arguments can be
used as a new method of proof for functional equations and
converse theorems for GL(3) and other higher rank groups.
This is joint work with Steve Miller.
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Ktypes of Peigendistributions
ChenBo Zhu, National University of Singapore, Singapore
We study representations of a classical group G which
admit certain Peigendistributions, where P is a parabolic
subgroup of G. Through examples, we shall explain how to
understand the Ktypes of Grepresentations generated by
these Peigendistributions.
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Conformal Geometry and
Global Solutions to the Yamabe Equations on Classical
PseudoRiemannian Manifolds
Toshiyuki Kobayashi, RIMS, Kyoto, Japan
The Yamabe operator on a pseudoRiemannian manifold is a
"modified" LaplaceBeltrami operator with scalar
curvature involved. The conformal group stabilizes the
space of global solutions to the Yamabe equation. Applying
this to the flat pseudoRiemannian manifold R^{p,q}, I
shall discuss analytic aspects of minimal representations
of indefinite orthogonal groups. This is a joint work with
B. Orsted.
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Estimates of Automorphic
forms and Representation Theory
Joseph Bernstein Tel Aviv University, Israel
Download/view PDF
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Regularity Theorems for
Automorphic Functionals
Joseph Bernstein Tel Aviv University, Israel
Download/view PDF
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Analytic Structures on
Representation Spaces of Reductive Groups
Joseph Bernstein Tel Aviv University, Israel
Download/view PDF
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Restriction of Unitary
Representations
Toshiyuki Kobayashi, RIMS, Kyoto, Japan
In this talk, I will discuss the restriction of unitary
representation of a real reductive Lie group G with respect
to its reductive subgroup H.
I will focus on the distinguished case where the
branching laws do not contain any continuous spectrum, with
some motivation, criterion, applications, and examples.
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Finite Dimensional
Representations of Invariant Differential Operators
Gerald Schwarz Brandeis University, USA
Let G be a reductive complex algebraic group and V
a finite dimensional Gmodule. Set B:=D(V)^{G},
the algebra of Ginvariant polynomial differential
operators on V. One can ask:
1) What is the representation theory of B? What
are the primitive ideals of B?
2) Does B have finite dimensional
representations? In particular, is B an FCRalgebra?
Little is known about these questions when G is
noncommutative. We give answers for the adjoint
representation of SL_{3}(C), already an
interesting and difficult case.
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Orbits and Invariants
Associated with a Pair of Commuting Involutions
Gerald Schwarz Brandeis University, USA
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On Zerofree Regions of
Lfunctions
Stephen Gelbart, Weizmann Institute of Science, Israel
There are (at least) three different approaches towards
reaching zerofree regions of the Riemann Zetafunction.
One is due to Hadamard and de la Vallee Pousin in the late
1890's, another to Ingham (in 1932) and Balasubramanian 
Ramachandra (1976), and the third to Selberg etc. (starting
about 1950).
We shall survey the proofs of each of these, and see
which can be used to generalize from the zetafunction to
arbitrary Lfunctions defined in terms of automorphic
cuspidal representations.
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Lifting of Covariants for
Nilpotent Orbits
Kyo Nishiyama, Kyoto University, Japan
It is known that the lifting of nilpotent orbits for a
reductive dual pair is very useful to study representations
of G and/or G'. In this talk, we
explain it briefly, and extend it to the lifting of some
equivariant coherent sheaves, which are regarded to be
modules of covariants. Under a reasonable condition, the
lifting will preserve multiplicities, hence we can get a
degree formula of the lifted sheaves modulo the geometric
degree of nilpotent orbits. (Joint work with Chenbo Zhu)
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Correspondence of Associated
Varieties of HarishChandra Modules
Kyo Nishiyama, Kyoto University, Japan
Let (G, G') be a reductive dual pair in the
stable range. We will give a description of good filtration
for the HarishChandra modules of G and G'
via the oscillator representation. As a consequence, we can
control the behavior of associated varieties under theta
correspondence. (Joint work with Chenbo Zhu)
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Multiplicity Free Spaces and
SchurWeylHowe Duality
Roe Goodman, Rutgers University, USA
The following topics will be covered in this set of
tutorials:
 Decompositions and Duality for Representations of
Reductive Groups
 Commutant Character Formula
 WeylSchur Duality and Frobenius Character Formula
 Tensor and Polynomial Invariants for Classical Groups
 Weyl Algebra and Howe Duality
 Spherical Harmonics and Duality
 Brauer Algebra and Harmonic Decomposition of Tensor
Spaces
 Seesaw Duality
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The Combinatorics of Multiplicity
Free Spaces
Friedrich Knop, Rutgers University, USA
We define a certain combinatorial structure for each
multiplicity free space. Then we show how that determines
the spectrum of invariant differential operators via
Capelli polynomials. Finally, we explain how to obtain the
Capelli polynomials as eigenfunctions of difference
operators.
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Special Functions and
Multiplicity Free Spaces
Friedrich Knop, Rutgers University, USA
Every multiplicity free space gives rise to an algebra
of difference operators. This algebra contains a copy of
sl_2 whose adjoint action integrates to an SL_2action. We
show how this permits the definition of various
multivariate generalizations of special functions, in
particular Laguerre, Meixner, and Bessel functions.
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MultiplicityFree Actions and
Tensor Algebras
EngChye Tan, National University of Singapore, Singapore
We shall use certain realizations of multiplicityfree
actions to study tensor product algebras. Let G be a
reductive group acting in a multiplicity free way on two
algebras A and B. We can form a
tensor product algebra by forming the algebra of covariants
of the tensor of these two algebras. It provides
information on the decomposition of tensor products of
representations coming from A and B.
We shall illustrate with G as the general linear group.
Download/view PDF
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G_{2}(q) Invariants
in Representations of D_{4}(q)
HungYean Loke, National University of Singapore,
Singapore
Download/view PDF
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Multiplicity Free Actions
Chal Benson and Gail Ratcliff, East Carolina University, USA
The action of a complex reductive group G on an
algebraic variety X is said to be multiplicity free when no
irreducible occurs more than once in the associated
representation of G on the ring of regular functions on X.
We will survey examples and applications for such actions,
which provide an organizational principle in Representation
Theory. Linear multiplicity free actions will be the main
focus in these lectures. Topics will include  examples of
multiplicity free decompositions,  criteria for linear
multiplicity free actions,  classification of linear
multiplicity free actions,  invariant polynomials and
polynomial coefficient differential operators, 
generalized binomial coefficients, and  spectra for
invariant polynomial coefficient differential operators.
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Invariant systems of
differential operators on a Hermitian symmetric space
Chal Benson, East Carolina University, USA
Let G/K be a Hermitian symmetric space. Left Ginvariant
systems of differential operators on G/K arise from Ad(K)invariant
subspaces of the complexified enveloping algebra for G.
This talk will report on recent joint work with D.
Buraczewski, E. Damek and G. Ratcliff concerning systems of
type (1,1) and their zeros.
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Multiplicity free actions and
analysis on the Heisenberg group
Gail Ratcliff, East Carolina University, USA
Multiplicity free actions occur in the study of Gelfand
pairs associated with compact extensions of the Heisenberg
group. Results from invariant theory give one information
on the spherical functions for such pairs, and the topology
of the Gelfand space.
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Vertex operators and McKay
correspondence
NaiHuan Jing North Carolina State University
In early 1980's McKay discovered that simply laced
Dynkin diagrams can be canonically and conceptually
obtained from Kleinian subgroups, the finite subgroups of
SL(2, C). The correspondence can be traced back to
Platonian solids dated 2000 years ago. McKay correspondence
has played a role in algebraic geometry (Nakajima's
revolutionary work) and Lie groups (eg. Steinberg, Kostant
and Vogan's works). We will give a new approach to the
correspondence based on affine Lie algebras and their
representations, in which the affine Dynkin diagrams are
manifested in the study. We show that the representation
theory of wreath products of a Kleinian subgroup and the
symmetric groups give naturally all the structures via
vertex operators. More interestingly our approach can be
easily generalized to quantum affine algebras as well as
some distinguished double covering groups of the
generalized symmetric groups based on ideas from affine Lie
algebras and vertex operator representations. The talk is
based on my work in 1989 and recent joint work with I.
Frenkel and W. Wang.
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