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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=Rn). Then the action of 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 Z determines a Clifford algebra Clif(Z). We write C=R1+Z and define a C-module as a Euclidean space X with an orthogonal action of Clif(Z), such that for every non-zero x in X, Cx is an invariant subspace. Then the unit ball in X + C can be made in a natural way into a hyperbolic space; a certain compactification of X + C gives the projective spaces and the sphere (which appears as a degenerate projective space). One can work with these without using any classification. One can also study "C-lines" and the collineation groups of the projective spaces.

 
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 Q-functions is an analogue of the Plancherel measure. Our aim is to calculate correlation functions of the shifted Schur measure in order to see a limit distribution of the measure.


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 GLn-orbits in an affine Grassmannian cone
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 GLn-orbits in the affine Grassmannian cone. The correspondence is related naturally to the actions of symplectic groups and orthogonal groups again on an affine cone of Grassmannian (as already implicitly pointed out by Ochiai). These actions in turn naturally comes from the adjoint action of a Levi subgroup on the nilpotent radical of parabolic subgroups.

 
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 × MM 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 QM for which Φ(Q) = Φ(M) holds. As an application one can give now symplectic proofs of all known convexity statements in Lie theory. As a prominent new example we will outline a symplectic proof of Kostant's non-linear convexity theorem.


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 GR - KC invariant subset C(S) of GC for each KC-orbit S on every flag manifold GC/P and conjectured that the connected component C(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 GR is of non-Hermitian type in my another work.

Recently the remaining problem for an arbitrary non-closed KC-orbit in Hermitian cases was solved. I want to talk in the seminar about this work by computing elementary examples. Thus the conjecture is completely solved affirmatively.

 
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 GLn
 
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),
and, for each 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),
where 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 φn, defined on H(n) by φn(x) = det Φ(x), is of positive type. The projective system (φn) 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, F4, E6, E7 and E8. 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 b-functions of prehomogeneous vector spaces with reducible representations. By examining the results of these calculations, we observe that b-functions of a large number of reducible prehomogeneous vector spaces have decompositions corresponding to those of representations. In this talk, we show that such phenomena can be ascribed to a certain multiplicity one property for group actions on polynomial rings. Our method can be applied equally to non-regular prehomogeneous vector spaces.

 
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 Sp2N(R),
G0 M0
X
K0 H0,
where both (G0, H0) and (K0, M0) form dual pairs, and both (G0, K0) and (M0, H0) are symmetric pairs.

Let ω be the Weil (oscillator) representation of Sp2N(R). Then we have the equality,

ω(U(g)K) = ω(U(m)H),
where g is the complexified Lie algebra of G0, K is the complexification of K, and U(g)K is the set of K-invariants of U(g).

When (G0, K0) is a symmetric pair of Hermitian type, we have already given the Capelli identities, which expresses particular elements of U(g)K by U(m)H in the image of ω.

In this talk, we give the Capelli identities, which conversely expresses particular elements of U(m)H by U(g)K for the see-saw pair called Case C:

U(p,q) U(r,s) × U(r,s)
X
Up × Uq U(r,s).
This is a joint work with Kyo Nishiyama.
 
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
 
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