2006 |
2005 |
**2004** |
2003 ]
## Lie Group and Representation Theory Seminar at RIMS 2004

List of speakers:

[
Herbert Heyer, Adam Koranyi (1), Adam Koranyi (2), Sho Matsumoto, Nishiyama Kyo, Eric Opdam, Bernhard Krötz, Hisayosi Matumoto, Toshihiko Matsuki, Toshiaki Hattori, Leticia Barchini, Birgit Speh, Jacques Faraut, Hung Yean Loke, Kazunari Sugiyama, Michel Duflo, Wachi Akihito, Dan Barbasch

Date: | April 13 (Tue), 2004, 16:30-17:30 | ||||||||||||||||

Room: | RIMS 402 | ||||||||||||||||

Speaker: | Herbert Heyer (Tuebingen, Germany) | ||||||||||||||||

Title: | Hecke pairs, generalized convolutions, and hypergroups | ||||||||||||||||

Abstract: [pdf] |
The talk is concerned with the notion of generalized translations
in locally
compact spaces introduced via convolution of measures. This concept has its
origin in the work of Frobenius on characters of groups, can be traced in the
theory of Hecke algebras, enjoyed a revival through the efforts of Delsarte
and Levitan in connection with Sturm-Liouville eigenvalue problems, and
reached the state of a useful axiomatization of hypergroups only about 30
years ago.
The speaker's aim is to describe the algebraic starting point of the notion of a hypergroup, to present a few striking examples arising from Gelfand pairs, and to expose some analytic aspects of the theory of locally compact hypergroups. Some of these aspects, notably the properties of the generalized Fourier transform of measures, enable the speaker to give an application to probability theory. | ||||||||||||||||

Date: | April 13 (Tue), 2004, 15:00-16:00 | ||||||||||||||||

Room: | RIMS 402 | ||||||||||||||||

Speaker: | Adam Koranyi (CUNY, USA) | ||||||||||||||||

Title: | LIOUVILLE-TYPE THEOREMS IN PARABOLIC GEOMETRY | ||||||||||||||||

Abstract: [pdf] |
G = O(n+1,1) acts on the n-sphere by conformal transformations.
In 1850 Liouville proved that, for n at least 3, any smooth conformal map
of an open subset of the sphere onto another one is the restriction of an
element of G.
In greater generality, let G be a simple real Lie group and P = MAN a
parabolic subgroup
(In the case of the n-sphere, M = O(n), A = R, N=R).
Then the action of ^{n}G on G/P is "multicontact" in the sense that it
preserves
a natural filtering of the tangent bundle induced by the root structure
(in the sphere-case the filtering is trivial).
It is also "conformal" in the sense that, in addition, the differential of
the action
at any point belongs to MA. In many cases (e. g. whenever P is non-maximal)
the analogue of Liouville's theorem holds for multicontact maps.
In almost all cases it holds for "conformal" maps.
A number of related results are known, most notably those proved by K.
Yamaguchi,
but the notion of multicontactness seems to be new.
A very simple proof, not using connections or classification,
will be given for the case of non-maximal P.
This is joint work with M. Cowling, F. De Mari and H. M. Reimann.
| ||||||||||||||||

Date: | April 16 (Fri), 2004, 10:30-11:30 | ||||||||||||||||

Room: | RIMS 402 | ||||||||||||||||

Speaker: | Adam Koranyi (CUNY, USA) | ||||||||||||||||

Title: | A SIMPLE DESCRIPTION OF THE SYMMETRIC SPACES OF RANK ONE | ||||||||||||||||

Abstract: [pdf] |
These are the hyperbolic spaces over R, C, H, O, the
corresponding four projective spaces,
and the sphere. It is usually difficult to make computations in them
because O is hard to handle;
the alternative way, using the structure theory of semisimple Lie groups,
is also relatively complicated.
Here a direct description of these spaces will be given, in which
everything is fairly easily computable.
A Euclidean space | ||||||||||||||||

Date: | June 1 (Tue), 2004, 17:00-18:00 | ||||||||||||||||

Room: | RIMS 402 | ||||||||||||||||

Speaker: | Sho Matsumoto ({ ) (Kyushu) | ||||||||||||||||

Title: | Measures on Young diagrams and symmetric functions | ||||||||||||||||

Abstract: [pdf] |
A limit distribution of the scaled first row in a Young diagram with respect to the Plancherel measure for symmetric groups is identical with that of
the scaled largest eigenvalue of a Hermitian matrix from the Gaussian
unitary ensemble
[Baik-Deift-Johansson, 1999].
This result is extended to the other rows in a Young diagram
by using correlation functions of the Plancherel measure
[Borodin-Okounkov-Olshanski, 2000].
The shifted Schur measure defined by Schur Prior to the seminar, Matumoto will give an introductory seminar from 15:30-16:30 at 402. | ||||||||||||||||

Date: | June 29 (Tue), 2004, 17:00-18:00 | ||||||||||||||||

Room: | RIMS 402 | ||||||||||||||||

Speaker: | Nishiyama Kyo (R ) (Kyoto Univ.) | ||||||||||||||||

Title: | Lifting of unimodular congruence classes of bilinear forms to the
GL-orbits in an affine Grassmannian cone_{n} | ||||||||||||||||

Abstract: [pdf] |
Recently Djokovic-Sekiguchi-Zhao and Ochiai are studying the unimodular
congruence classes of bilinear forms.
The invariant ring of the unimodular action on the space of bilinear forms is known to be a polynomial ring, which means the affine categorical quotient is an affine space. In spite of it, one of the results of DSZ tells us that the null cone contains infinite number of orbits, which are not separate by invariants. While Ochiai proved that the nilpotent orbits in the null cone can be classified inductively.
In this talk, we consider a correspondence between the unimodular congruence
classes and certain | ||||||||||||||||

Date: | September 14 (Tues), 2004, 16:30-17:30 | ||||||||||||||||

Room: | RIMS 402 | ||||||||||||||||

Speaker: | Eric Opdam (Amsterdam and RIMS) | ||||||||||||||||

Title: | Harmonic analysis for affine Hecke algebras | ||||||||||||||||

Abstract: [pdf] |
The study of the Plancherel decomposition of
affine Hecke algebras is motivated by its role in the
representation theory of p-adic reductive groups. I will give
an overview of results concerning the Plancherel measure,
the Schwartz algebra and the analytic R-groups. Then I will
discuss some natural conjectures arising from this picture.
| ||||||||||||||||

Date: | October 12 (Tue), 2004, 16:30-17:30 | ||||||||||||||||

Room: | RIMS 402 | ||||||||||||||||

Speaker: | Bernhard Krötz (RIMS) | ||||||||||||||||

Title: | Lagrangian submanifolds and moment convexity | ||||||||||||||||

Abstract: [pdf] |
Consider a Hamiltonian torus action T × M → M on a
compact and connected symplectic manifold M. Associated to this
data is the moment map Φ: M → t*. It is a remarkable
structural fact, due to Atiyah and Guillemin-Sternberg, that
the image of Φ is a convex polytope. The AGS-theorem
was generalized by Duistermaat who showed that if Q is
Lagrangian submanifold of M which arises as the fixed point
set of a T-compatible anti-symplectic involution, then
Φ(Q) = Φ(M) is a convex polytope.
In this talk we present a result which extends Duistermaat's
Theorem in the sense that it substantially enlarges the class of Lagrangians
Prior to this seminar, Krötz will give an introductory lecture on Hamiltonian torus actions from 15:00-16:00 in the same room. | ||||||||||||||||

Date: | October 22 (Fri), 2004, 17:00-18:00 | ||||||||||||||||

Room: | RIMS 402 | ||||||||||||||||

Speaker: | Hisayosi Matumoto ({v`) (University of Tokyo) | ||||||||||||||||

Title: | Derived functor modules arising as large irreducible constituents of degenerate principal series (joint work with Peter E. Trapa) | ||||||||||||||||

Abstract: [pdf] |
We consider a degenerate principal series of G = Sp (p,q) and SO*(2n) with an infinitesimal character
appearing as a weight of some finite-dimensional G-representation.
We prove that each irreducible constituent of the maximal
Gelfand-Kirillov dimension is a derived functor module.
We also show at a most singular parameter each irreducible constituent
is weakly unipotent and unitarizable.
Moreover, any weakly unipotent representation associated to a real form of the
corresponding Richardson orbit is unique up to isomorphism and can be
embedded into a degenerate principal series of the most singular
integral parameter, except for the very even cases.
We also discuss edge-of-wedge-type embeddings of derived functor
modules into degenerate principal series.
Prior to this seminar, Matumoto will give an introductory lecture on unipotent representations from 15:00-16:30 in the same room. | ||||||||||||||||

Date: | October 26 (Tue), 2004, 16:30-17:30 | ||||||||||||||||

Room: | RIMS 402 | ||||||||||||||||

Speaker: | Toshihiko Matsuki (ؕqF) (Kyoto University) | ||||||||||||||||

Title: | Equivalence of domains arising from duality of orbits on flag manifolds III | ||||||||||||||||

Abstract: [pdf] |
In my joint work with Gindikin, we defined a G invariant subset _{R} - K_{C}C(S)
of G for each _{C}K-orbit _{C}S on every flag manifold G and conjectured
that the connected component _{C}/PC(S)_{0} of the identity would be equal to the
Akhiezer-Gindikin domain D if S is of nonholomorphic type. This conjecture
was proved for closed S by the works of Wolf-Zierau (Hermitian cases) and
Fels-Huckleberry (non-Hermitian cases). For open S it was proved in my work
generalizing the result of Barchini. (This work also gave an alternative
proof for closed S in non-Hermitian cases.) It was also proved for all the
other orbits when G is of non-Hermitian type in my another work.
_{R}
Recently the remaining problem for an arbitrary non-closed | ||||||||||||||||

Date: | November 2 (Tue), 2004, 16:30-17:30 | ||||||||||||||||

Room: | RIMS Room 402 | ||||||||||||||||

Speaker: | Toshiaki Hattori (r) (T.I.T.) | ||||||||||||||||

Title: | On essential spectrum of manifolds with ends | ||||||||||||||||

Abstract: [pdf] |
We find a sufficient condition written in a geometric language for the existence of bands of essential spectrum of complete noncompact Riemannian manifolds and consider the lower bound of the essential spectrum. By using them, we recover some of the known results for locally symmetric spaces of finite volume and treat the complete manifolds of infinite volume obtained from manifolds with corners. | ||||||||||||||||

Date: | November 22 (Mon), 2004, 16:30-17:30 | ||||||||||||||||

Room: | RIMS Room 005 | ||||||||||||||||

Speaker: | Leticia Barchini (Oklahoma State Univesity) | ||||||||||||||||

Title: | Positivity of zeta distributions and small representations | ||||||||||||||||

Abstract: |
The purpose of the talk is (1) to describe a theory analogous to that of Riesz distributions and Wallach set in the setting of non-Euclidean Jordan algebras. We show how these "Riesz distributions" play a role in giving unitary realization of some irreducible representations. (2) to describe some invariants of the resulting representation and describe their space of smooth Whittaker vectors. This talk is partially based on work with Sepanski-Zierau and in part based on work in progress with Binegar and Zierau. | ||||||||||||||||

Date: | November 26 (Fri), 2004, 16:30-17:30 | ||||||||||||||||

Room: | RIMS Room 402 | ||||||||||||||||

Speaker: | Birgit Speh (Cornell University) | ||||||||||||||||

Title: | Convergence of the spectral side of the Arthur Selberg trace formula | ||||||||||||||||

Abstract: [pdf] |
The Arthur trace formula is an identity between distributions
indexed by spectral data on one side and
geometric data on the other side.
On the spectral side this leads
to an integral-series that is only known to
converge conditionally. The absolute
convergence has been reduced by W. Müller
to a problem about local components of
automorphic representations.
I will discuss these local problems and show how they can be solved for
GL
_{n} | ||||||||||||||||

Date: | December 6 (Mon), 2004, 16:30-17:30 | ||||||||||||||||

Room: | RIMS Room 005 (underground) | ||||||||||||||||

Speaker: | Jacques Faraut (Paris) | ||||||||||||||||

Title: | Infinite dimensional harmonic analysis and Polya functions | ||||||||||||||||

Abstract: [pdf] |
Spherical pairs, which have been introduced by Olshanski, are
inductive limits of Gelfand pairs.
For such a pair (G,K),
G = ∪_{n=1}^{∞} G(n),
K = ∪_{n=1}^{∞} K(n),
G(n) ⊂ G(n+1),
K(n) = G(n) ∩ K(n+1),
n, ((G(n),K(n)) is a Gelfand pair. For some
spherical pairs, the spherical functions are characterized by a
multiplicative property, and a class of one variable functions comes in the
theory. A basic example is the space of infinite dimensional Hermitian
matrices
H(∞) = ∪_{n=1}^{∞} H(n),
H(n) is the space of n × n Hermitian matrices, for which
K(n) = U(n), the unitary group, and G(n) = U(n) \ltimes H(n), the
corresponding motion group.
A continuous function Φ on R is said to be a Pólya
function if
Φ(0) = 1, and if, for every n,
the function φ_{n}, defined on H(n) by φ_{n}(x) = det Φ(x),
is of positive type. The projective system (φ_{n})
defines a function φ on H(∞); this function φ
is spherical, and all spherical functions are obtained in that way. The
Pólya functions have been determined by Olshanski and Vershik, and also
by Pickrell. Surprinsingly, this class of functions has been considered a
long time ago by Pólya and Schoenberg in a very different setting.
| ||||||||||||||||

Date: | December 14 (Tue), 2004, 16:30-17:30 | ||||||||||||||||

Room: | RIMS Room 402 | ||||||||||||||||

Speaker: | Hung Yean Loke (Singapore National University) | ||||||||||||||||

Title: | Exceptional dual pair correspondences of complex groups | ||||||||||||||||

Abstract: [pdf] |
In this talk I will discuss an on-going project to investigate dual
pairs correspondences of the minimal representations of the complex
exceptional groups, F_{4}, E_{6}, E_{7} and E_{8}. The calculations relies on
some small Verma modules constructed by Gross and Wallach and the
naturality of the Zuckerman functors.
| ||||||||||||||||

Date: | January 20 (Thu), 2005, 17:00-18:00 | ||||||||||||||||

Room: | RIMS Room 005 | ||||||||||||||||

Speaker: | Kazunari Sugiyama (Ra) | ||||||||||||||||

Title: | Multiplicity One Property and the Decomposition of b-Functions | ||||||||||||||||

Abstract: [pdf] |
This talk is based on a joint work with Professor Fumihiro Sato
(Rikkyo University).
Recently, extensive calculations have been made on | ||||||||||||||||

Date: | February 23 (Wed), 2005, 16:30-17:30 | ||||||||||||||||

Room: | RIMS Room 202 | ||||||||||||||||

Speaker: | Michel Duflo (Paris) | ||||||||||||||||

Title: | Restrictions of discrete series of semisimple Lie groups | ||||||||||||||||

Abstract: [pdf] |
In this lecture, I will recall first the classification of discrete series representations of real algebraic Lie group, in the setting of the orbit method, and discuss related properties of Lie algebras. In the case of reductive groups, I will present some results on their restrictions to closed subgroup. | ||||||||||||||||

Date: | February 24 (Thu), 2005, 11:00-12:00 | ||||||||||||||||

Room: | RIMS Room 005 | ||||||||||||||||

Speaker: | Wachi Akihito (anPm) (Hokkaido Institute of Technology) | ||||||||||||||||

Title: | Capelli identities for symmetric pairs of non-Hermitian type | ||||||||||||||||

Abstract: [pdf] |
Consider a see-saw pair of real reductive Lie groups
in the real symplectic group Sp_{2N}(R),
G_{0}, H_{0}) and (K_{0}, M_{0}) form dual pairs,
and both (G_{0}, K_{0}) and (M_{0}, H_{0}) are symmetric pairs.
Let ω be the Weil (oscillator) representation of U(g)^{K}) = ω(U(m)^{H}),
g is the complexified Lie algebra of G_{0},
K is the complexification of K,
and U(g)^{K} is the set of K-invariants of U(g).
When (
In this talk, we give the Capelli identities,
which conversely expresses particular elements of
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Date: | March 18 (Fri), 2005, 18:00-19:00 | ||||||||||||||||

Room: | RIMS Room 402 | ||||||||||||||||

Speaker: | Dan Barbasch (Cornell University) | ||||||||||||||||

Title: | Relevant and petite K-types for real and p-adic groups
| ||||||||||||||||

Abstract: [pdf] |
The unitary dual of a reductive group over a local field plays an important
role in noncommutative harmonic analysis. Its structure is also relevant for
many problems in analysis, mathematical physics and automorphic forms.
In this talk I will survey progress on the determination of the unitary
dual. In particular, relevant W-types are a set of representations of the
Weyl group which give necessary and sufficient conditions for determining
the spherical unitary dual of a split p-adic group. Petite K-types are
represntations of the maximal compact subgroup of a real group which are
closely related to the relevant W-types. They provide a means to transfer
results about the unitary dual of p-adic groups to the case of various series
of representations of real groups.
| ||||||||||||||||

Organizer: | Toshiyuki Kobayashi | ||||||||||||||||

© Toshiyuki Kobayashi