Lie Group and Representation Theory Seminar, Kyoto 2004
Lie Group and Representation Theory Seminar, Kyoto 2004 | Date: | December 14 (Tue), 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: | December 6 (Mon) 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 $(G,K)$, $$G=\bigcup _{n=1}^{\infty } G(n),\quad K=\bigcup _{n=1}^{\infty } K(n), \quad G(n)\subset G(n+1),\quad K(n)=G(n)\cap K(n+1),$$ and, for each $n$, $\bigl((G(n),K(n)\bigr)$ 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 $$H(\infty )=\bigcup _{n=1}^{\infty } H(n),$$ where $H(n)$ is the space of $n\times n$ 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 $\Phi $ on $\mathbb{R}$ is said to be a P\'olya function if $\Phi (0)=1$, and if, for every $n$, the function $\varphi _n$, defined on $H(n)$ by $\varphi _n(x)=\det \Phi (x)$, is of positive type. The projective system $(\varphi _n)$ defines a function $\varphi $ on $H(\infty )$; this function $\varphi $ is spherical, and all spherical functions are obtained in that way. The P\'olya 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 P\'olya and Schoenberg in a very different setting. |
| Date: | November 26 (Fri) 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: | November 22 (Mon) 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 2 (Tue) 16:30-17:30 |
| Room: | RIMS Room 402 |
| Speaker: | rΊ Toshiaki Hattori (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: | October 26 (Tue) 16:30-17:30 |
| Room: | RIMS 402 |
| Speaker: | ΌΨqF Toshihiko Matsuki (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_R-K_C invariant
subset C(S)
of G_C for each K_C-orbit S on every flag manifold G_C/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 G_R is of non-Hermitian type in my another work.
Recently the remaining problem for an arbitrary non-closed K_C-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: | October 22 (Fri), 2004, 10:30-11:30 |
| Room: | RIMS 402 |
| Speaker: | Bernhard Krötz (RIMS) |
| Title: | Working Seminar on Integral Geometry 3 |
| Abstract: [pdf] |
We will finish our discussion on the space for horocycles with no real points attached to a compact symmetric space. A key fact in this context is a certain uniform boundedness result for matrix coefficients (due to Clerc) which will be explained in detail. |
| Date: | October 22 (Fri), 2004, 17:00-18:00 |
| Room: | RIMS 402 |
| Speaker: | Ό{v` Hisayosi Matumoto (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
$\SO^\ast(2n)$ 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 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 $T\times M\to M$ on a
compact and connected symplectic manifold $M$. Associated to this
data is the moment map $\Phi: M\to {\mathfrak t}^*$. It is a remarkable
structural fact, due to Atiyah and Guillemin-Sternberg, that
the image of $\Phi$ 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
$\Phi(Q)=\Phi(M)$ 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 $Q\subset M$ for which $\Phi(Q)=\Phi(M)$ 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, Kroetz will give an introductory lecture on Hamiltonian torus actions from 15:00-16:00 in the same room. |
| 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: | June 29 (Tue), 2004, 17:00-18:00 |
| Room: | RIMS 402 |
| Speaker: | ΌR@ iNishiyama Kyo) (Kyoto Univ.) |
| Title: | Lifting of unimodular congruence classes of bilinear forms to the $ GL_n$-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 $ GL_n $-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: | June 1 (Tue), 2004, 17:00-18:00 |
| Room: | RIMS 402 |
| Speaker: | Ό{ Ω@@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: | 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: | 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^n). 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 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 ELiouville 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: | March 9 (Tue), 2004, 16:00-17:00 |
| Room: | RIMS 402 |
| Speaker: | Munibur R. Chowdhury (the University of Dhaka, Bangladesh) |
| Title: | Arthur Cayley and his contribution to abstract group theory |
| Abstract: [pdf] |
We critically reexamine in considerable detail Cayley's first three papers on group theory (1854- 59), with special reference to his formulation of the (abstract) group concept. We show convincingly (we hope) that Cayley, writing his first paper on November 2, 1853, was in full and conscious possession of the abstract group concept, and that - as far as finite groups are concerned - his definition was complete and unequivocal, refuting opinion expressed by some earlier writers. Already in the first paper Cayley classified the abstract groups of orders upto 6, and suggested that there might exist composite numbers n such that the only abstract group of order n is the cyclic group of that order. We also discuss Cayley's motivation for generalizing the then current concept of a permutation group. Cayley extended the classification to groups of order 8 in the third paper. There he also initiated the study of groups in terms of generators and relations (a procedure usually attributed to Walter Dyck), and in this way constructed the abstract dihedral group of order 2n. However, these pioneering studies were swept away by the then burgeoning surge of permutation groups, and apparently went completed unheeded by his contemporaries. |
| Date: | February 20 (Fri), 2004, 16:00--17:00 |
| Room: | RIMS 402 |
| Speaker: | Eric Sommers (University of Massachusetts, Amherst) |
| Title: | Functions on nilpotent orbits and their covers |
| Abstract: |
Extending work of McGovern and Graham, we conjecture a formula for the functions on covers of nilpotent orbits and prove it in many cases. One application of this result is an explicit computation for the classical groups of a part of the Lusztig-Vogan bijection (this is a bijection between the set of irreducible equivariant coherent sheaves on nilpotent orbits in a simple Lie algebra and the set consisting of its irreducible finite-dimensional representations). |
| Date: | February 17 (Tue), 2004, 17:00-18:00 |
| Room: | RIMS 402 |
| Speaker: | Raphael Rouquier (Paris VII) |
| Title: | Categorification of Weyl groups and Lie algebras |
| Abstract: |
It is classical that various actions of Weyl groups or Lie algebras on vector spaces come from functors acting on abelian or triangulated categories of algebraic or geometric origin, whose Grothendieck group is that space. We want to explain that the natural transformations between these functors should satisfy certain algebraic relations, leading to a better control of the triangulated categories acted on. We give two precise such frameworks. For Weyl groups, we use Soergel bimodules. In a joint work with Chuang, we explain the setting for sl2, which leads to a construction of equivalences of derived categories between blocks of Hecke algebras of type A. |
| Date: | February 10 (Tue) , 2004, 17:00-18:00 |
| Room: | RIMS 402 |
| Speaker: | Soo Teck Lee (NUS, Singapore) |
| Title: | A basis for the k-fold tensor product algebra of GL_n(C) |
| Abstract: [pdf] |
| Date: | January 13 (Tue), 2004, 16:00-17:00 |
| Room: | RIMS 402 |
| Speaker: | Andreas Nilsson (Royal Institute of Technology, Stockholm, Sweden) |
| Title: | Investigation of L^p-boundedness for certain multipliers |
| Abstract: |
Together with professor Kobayashi, I have been trying to characterize multipliers by group actions. To begin with we were only concerned with the characterization and hence only asked for L^2-boundedness. But it is natural to ask which of the multipliers that are bounded on L^p as well. The tools used include deLeeuw's theorem on restriction to affine subspaces and Fefferman's ball multiplier theorem. |
| Date: | January 9 (Fri), 2004, 16:00-17:00 |
| Room: | RIMS 402 |
| Speaker: | Andreas Nilsson (Royal Institute of Technology, Stockholm, Sweden) |
| Title: | Some Theorems of deLeeuw on multipliers |
| Abstract: [pdf] |
To check L^p-boundedness of multipliers it is sometimes useful to be able to restrict to simpler situations. deLeeuw has proved that if we restrict an L^p-bounded multiplier operator to a linear subspace then the resulting multiplier operator will also be bounded. I will talk about the proof of this and some related results. In my second talk I will give some applications to this. |