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Lie Groups and Representation Theory Seminar 2009

List of speakers:
Eric Opdam (#1), Erik van den Ban, Patrick Delorme, Mogens Flensted-Jensen, Masaki Kashiwara, Toshiyuki Kobayashi, Toshihiko Matsuki, Hisayosi Matumoto, Hiroyuki Ochiai, Yasunori Okada, Eric Opdam (#2), Toshio Oshima, Yoshiki Otobe, Hideko Sekiguchi, Jiro Sekiguchi, Nobukazu Shimeno, Gombodorj Bayarmagnai, Bernhard Krötz, Salah Mehdi, Peter Trapa, Roger Zierau, Sigurdur Helgason, Fulton Gonzalez, Angela Pasquale, Henrik Schlihitkrull, Michel Duflo, Ryosuke Kodera, Shunsuke Tsuchioka, Gert Heckman #1, Gert Heckman #2, Hiroshi Iritani, Carlos Olmos, Michael Pevzner, Daniel Sternheimer, Takaaki Nomura, Toshikazu Sunada #1, Toshikazu Sunada #2, Kyo Nishiyama,
(GCOE Lectures)
Date and place:
  1. January 8 (Thu), 2009, 17:00-18:00, Room 123
  2. January 9 (Fri), 2009, 17:00-18:00, Room 123
  3. January 22 (Thu), 2009, 17:00-18:00, Room 370
  4. January 23 (Fri), 2009, 17:00-18:00, Room 370
Speaker: Eric Opdam (University of Amsterdam)
Title: The spectral category of Hecke algebras and applications
Lecture 1: Reductive p-adic groups and Hecke algebras.
Lecture 2: Affine Hecke algebras and harmonic analysis.
Lecture 3: The spectral category and correspondences of tempered representations.
Lecture 4: Example: Lusztig's unipotent representations for classical groups.
[ pdf ]
Hecke algebras play an important role in the harmonic analysis of a p-adic reductive group. On the other hand, their representation theory and harmonic analysis can be described almost completely explicitly. This makes affine Hecke algebras an ideal tool to study the harmonic analysis of p-adic groups. We will illustrate this in this series of lectures by explaining how various components of the Bernstein center contribute to the level-0 L-packets of tempered representations, purely from the point of view of harmonic analysis.

We define a "spectral category" of (affine) Hecke algebras. The morphisms in this category are not algebra morphisms but are affine morphisms between the associated tori of unramified characters, which are compatible with respect to the so-called Harish-Chandra μ-functions. We show that such a morphism generates a Plancherel measure preserving correspondence between the tempered spectra of the two Hecke algebras involved. We will discuss typical examples of spectral morphisms.

We apply the spectral correspondences of affine Hecke algebras to level-0 representations of a quasi-split simple p-adic group. We will concentrate on the example of the special orthogonal groups SO2n+1(K). We show that all affine Hecke algebras which arise in this context admit a *unique* spectral morphism to the Iwahori-Matsumoto Hecke algebra, a remarkable phenomenon that is crucial for this method. We will recover in this way Lusztig's classification of "unipotent" representations.

(Conference in honor of Toshio Oshima's 60th birthday "Differential Equations and Symmetric Spaces")
Date: 13-16 January, 2009
Place: Graduate School of Mathematical Sciences, the University of Tokyo
Speakers:Erik van den Ban (Utrecht University)
Patrick Delorme (University of Aix-Marseille II)
Mogens Flensted-Jensen (University of Copenhagen)
Masaki Kashiwara (柏原正樹) (RIMS)
Toshiyuki Kobayashi (小林俊行) (The University of Tokyo)
Toshihiko Matsuki (松木敏彦) (Kyoto University)
Hisayosi Matumoto (松本久義) (The University of Tokyo)
Hiroyuki Ochiai (落合啓之) (Nagoya University)
Yasunori Okada (岡田靖則) (Chiba University)
Eric Opdam (University of Amsterdam)
Toshio Oshima (大島利雄) (The University of Tokyo)
Yoshiki Otobe (乙部厳己) (Shinshu University)
Hideko Sekiguchi (関口英子) (The University of Tokyo)
Jiro Sekiguchi (関口次郎) (Tokyo University of Agriculture and Technology)
Nobukazu Shimeno (示野信一) (Okayama University of Science)
Date: February 3 (Tue), 2009, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Gombodorj Bayarmagnai (University of Tokyo)
Title: The (g,K)-module structure of principal series and related Whittaker functions of SU(2,2)
[ pdf ]
In this talk the basic object will be the principal series representataion of SU(2,2), parabolically induced by the minimal parabolic subgroup. We discuss about the (g,K)-module structure on that type of principal series explicitely, and provide various integral expressions of some smooth Whittaker functions with certain K-types.
(GCOE Spring School on Representation Theory)
Date: March 12-17, 2009
Place: Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Bernhard Krötz (Germany)
Title: Harish-Chandra modules
[ pdf ]
We plan to give a course on the various types of topological globalizations of Harish-Chandra modules. It is intended to cover the following topics:
  1. Topological representation theory on various types of locally convex vector spaces.
  2. Basic algebraic theory of Harish-Chandra modules
  3. Unique globalization versus lower bounds for matrix coefficients
  4. Dirac type sequences for representations
  5. Deformation theory of Harish-Chandra modules
The new material presented was obtained in collaboration with Joseph Bernstein and Henrik Schlichtkrull. A first reference is the recent preprint "Smooth Frechet Globalizations of Harish-Chandra Modules" by J. Bernstein and myself, downloadable at arXiv:0812.1684v1.
Speaker: Salah Mehdi (France)
Title: Enright-Varadarajan modules and harmonic spinors
[ pdf ]
The aim of these lectures is twofold. First we would like to describe the construction of the Enright-Varadarajan modules which provide a nice algebraic characterization of discrete series representations. This construction uses several important tools of representations theory. Then we shall use the Enright-Varadarajan modules to define a product for harmonic spinors on homogeneous spaces.
Speaker: Peter Trapa (USA)
Title: Special unipotent representations of real reductive groups
[ pdf ]
These lectures are aimed at beginning graduate students interested in the representation theory of real Lie groups. A familiarity with the theory of compact Lie groups and the basics of Harish-Chandra modules will be assumed. The goal of the lecture series is to give an exposition (with many examples) of the algebraic and geometric theory of special unipotent representations. These representations are of considerable interest; in particular, they are predicted to be the building blocks of all representation which can contribute to spaces of automorphic forms. They admit many beautiful characterizations, but their construction and unitarizability still remain mysterious.

The following topics are planned:

  1. Algebraic definition of special unipotent representations and examples.
  2. Localization and duality for Harish-Chandra modules.
  3. Geometric definition of special unipotent representations.
Speaker: Roger Zierau (USA)
Title: Dirac Cohomology
[ pdf ]
Dirac operators have played an important role in representation theory. An early example is the construction of discrete series representations as spaces of L2 harmonic spinors on symmetric spaces G/K. More recently a very natural Dirac operator has been discovered by Kostant; it is referred to as the cubic Dirac operator. There are algebraic and geometric versions. Suppose G/H is a reductive homogeneous space and $\mathfrak g = \mathfrak h + \mathfrak q$. Let S\mathfrak q be the restriction of the spin representation of SO(\mathfrak q) to HSO(\mathfrak q). The algebraic cubic Dirac operator is an H-homomorphism \mathcal D: V \otimes S\mathfrak qV \otimes S\mathfrak q, where V is an $\mathfrak g$-module. The geometric geometric version is a differential operator acting on smooth sections of vector bundles of spinors on G/H. The algebraic cubic Dirac operator leads to a notion of Dirac cohomology, generalizing $\mathfrak n$-cohomology. The lectures will roughly contain the following.
  1. Construction of the spin representations of \widetilde{SO}(n).
  2. The algebraic cubic Dirac operator \mathcal D: V \otimes S\mathfrak qV \otimes S\mathfrak q will be defined and some properties, including a formula for the square, will be given.
  3. Of special interest is the case when H=K, a maximal compact subgroup of G and V is a unitarizable $(\mathfrak g,K)$-module. This case will be discussed.
  4. The Dirac cohomology of a finite dimensional representation will be computed. We will see how this is related to $\mathfrak n$-cohomology of V.
  5. The relationship between the algebraic and geometric cubic Dirac operators will be described. A couple of open questions will then be discussed.
The lectures will be fairly elementary.
Contact: Toshio Oshima [ Email ] and Toshiyuki Kobayashi [ Email ]
Workshop on Integral Geometry and Group Representations
Date: August 5 (Wed)-10 (Mon), 2009
Place: Tambara Institute of Mathematical Sciences, The University of Tokyo (東京大学玉原国際セミナーハウス)
Invited speakers include:
Mike Eastwood, Sigurdur Helgason, Angela Pasquale and Henrik Schlichtkrull
Organizers: Fulton B. Gonzalez, Tomoyuki Kakehi, Toshiyuki Kobayashi and Toshio Oshima
Mini-Conference "Integral Geometry and Representation Theory" [ pdf ]
Date: August 12 (Wed), 2009
Place: Room 002, Graduate School of Mathematical Sciences, the University of Tokyo

Speaker: Sigurdur Helgason (MIT)
Title: Radon transforms and some applications

Speaker: Fulton Gonzalez (Tufts University)
Title: Multitemporal wave equations: mean value solutions

Speaker: Angela Pasquale (Universite Metz)
Title: Analytic continuation of the resolvent of the Laplacian in the Euclidean setting
Abstract: We discuss the analytic continuation of the resolvent of the Laplace operator on symmetric spaces of the Euclidean type and some generalizations to the rational Dunkl setting.

Speaker: Henrik Schlihitkrull (Univ. of Copenhagen)
Title: Decay of smooth vectors for regular representation
Abstract: Let G/H be a homogeneous space of a Lie group, and consider the regular representation L of G on E=L^p(G/H). A smooth vector for L is a function f in E such that g mapsto L(g) f is smooth, G to E. We investigate circumstances under which all such functions decay at infinity (jt with B. Krötz).
(GCOE Lectures)
Date: October 7 (Wed) & 9 (Fri), 2009, 16:30-17:30
Place: Room 128, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Michel Duflo (Paris 7)
Title: Lecture 1. Representations of classical Lie super-algebras
Lecture 2. Associated varieties for Representations of classical Lie super-algebras
[ pdf ]
(Lecture 1)
In this lecture, I'll survey classical topics on finite dimensional representations of classical Lie super-algebras, in particular the notion of the degree of atypicality.

(Lecture 2)
In this lecture, I'll discuss the notion of "Associated varieties for Representations of classical Lie super-algebras (joint work with Vera Serganova)" and the relation with the degree of atypicality. This is related to a conjecture of Kac and Wakimoto.

Date: October 13 (Tue), 2009, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Ryosuke Kodera (小寺諒介) (University of Tokyo)
Title: Extensions between finite-dimensional simple modules over a generalized current Lie algebra
[ pdf ]
$\mathfrak{g}$を$\mathbb{C}$上の有限次元半単純Lie代数,$A$を有限生成可換$\mathbb {C}$代数とする. テンソル積$A \otimes \mathfrak{g}$に自然にLie代数の構造を与えたものを一般化されたカレントLie代数と呼ぶ. 一般化されたカレントLie代数の任意の2つの有限次元既約表現に対して,その1次のExt群を完全に決定することができたので,その結果について発表する.
Date: October 15 (Thu), 2009, 16:30-18:00
Place: Room 122, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Shunsuke Tsuchioka (土岡俊介) (RIMS, Kyoto University)
Title: Hecke-Clifford superalgebras and crystals of type $D^{(2)}_{l}$
[ pdf ]
It is known that we can sometimes describe the representation theory of "Hecke algebra" by "Lie theory". Famous examples that involve the Lie theory of type $A^{(1)}_n$ are Lascoux-Leclerc-Thibon's interpretation of Kleshchev's modular branching rule for the symmetric groups and Ariki's theorem generalizing Lascoux-Leclerc-Thibon's conjecture for the Iwahori-Hecke algebras of type A.

Brundan and Kleshchev showed that some parts of the representation theory of the affine Hecke-Clifford superalgebras and its finite-dimensional "cyclotomic" quotients are controlled by the Lie theory of type $A^{(2)}_{2l}$ when the quantum parameter $q$ is a primitive $(2l+1)$-th root of unity. In this talk, we show that similar theorems hold when $q$ is a primitive $4l$-th root of unity by replacing the Lie theory of type $A^{(2)}_{2l}$ with that of type $D^{(2)}_{l}$.

Date: November 4 (Wed), 2009, 16:30-18:00
Place: Room 128, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Gert Heckman (IMAPP, Faculty of Science, Radboud University Nijmegen)
Title: Birational hyperbolic geometry
[ pdf ]
We study compactifications for complex ball quotients. We first recall the Satake-Bailey-Borel compactification and the Mumford resolution. Then we discuss compactifications of ball quotients minus a totally geodesic divisor. These compactifications turn up for a suitable class of period maps (related to the lecture of the speaker on friday on "Geometric structures").
IPMU workshop "Quantizations, integrable systems and representation theory"
Date: November 5 (Thu)-6 (Fri), 2009
Place: Seminar Room at IPMU Prefab. B, The University of Tokyo (Kashiwa Campus)
Invited speakers: Gert Heckman (IMAPP, Faculty of Science, Radboud University Nijmegen)
Hiroshi Iritani (Kyushu University)
Carlos Olmos (National University of Cordoba, Argentina)
Michael Pevzner (Univ. de Reims)
Daniel Sternheimer* (Univ. de Bourgogne)
   *to be confirmed
Organizers: Martin Guest, Toshiyuki Kobayashi, Toshitake Kohno
Tentative Program: November 5
10:30-12:00   Pevzner, Composition formulas seen through representation theory I
13:30-15:00   Olmos, Submanifolds and holonomy
15:00-15:30   Tea
15:30-17:00   Heckman, Complex hyperbolic structures
Evening   workshop dinner
November 6
10:30-12:00   Pevzner, Composition formulas seen through representation theory II
13:30-15:00   Iritani, Quantization in semi-infinite Hodge theory
15:00-15:30   Tea
15:30-17:00   Sternheimer*
(集中講義:数理科学特別講義 I)[ レポート問題 ]
Date: December 7 (Mon)-11 (Fri), 2009, 14:40-16:40
Place: Room 123, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Takaaki Nomura (野村隆昭) (Kyushu University)
Title: 等質錐と等質ジーゲル領域
[ pdf ]
本講義は,E. Cartanの問題「$n \geqq 4$ のとき,$\C^n$に非対称な等質有界領域が存在するか」に肯定的な解答を与えた等質ジーゲル領域($\C$の上半平面の高次元化),及び等質開凸錐に関するものである.私や名古屋大学の伊師英之氏,金沢大学の甲斐千舟氏の近年の結果(共同あるいは私や彼ら自身の単独研究),特に対称領域の特徴付けに関する結果の解説を中心に講義をするが,そこで用いられる基本的な概念や代数的構造物については,例と共に詳しく述べる.定理についても,完全な証明を与えることより,むしろ例を通して理解するというような方針で進めたい.


  1. 正規$j$代数(Piatetski-Shapiro代数)
    — ジーゲル領域に単純推移的にアフィン変換で作用する分裂可解リー群のリー代数
  2. (ユークリッド型及び複素半単純)ジョルダン代数
  3. (エルミート型)ジョルダン3重系
  4. Vinberg の意味でのクラン(コンパクト正規左対称代数)
    — 以上の(1)〜(4)は,等質領域の接空間に定義される代数的構造物ということで捉えることができる.相互の関係についても述べる.
  5. 等質錐に付随する擬逆元写像
    — 行列にその逆行列を対応させるという写像の一般化
  6. 等質ジーゲル領域のケイリー変換
    — 上半平面を単位円の内部に写す変換$z \mapsto \frac{z-i}{z+i}$の高次元化で,(5)の擬逆元写像を用いて定義される.対称領域の場合は,パラメタを適当に選べば,Korány-Wolf によって導入された,エルミート対称空間のHarish-Chandra モデルを,その非有界モデルであるジーゲル領域に写す変換の逆変換に等しくなる.
  7. 等質錐に付随する基本相対不変式
    — 行列の首座小行列式の一般化にあたる
  8. ジョルダン代数の表現と準対称ジーゲル領域
(集中講義:数理科学特別講義 IX)
Date: December 14 (Mon)-18 (Fri), 2009, 14:40-16:40
Place: Room 123, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Toshikazu Sunada (砂田利一) (Meiji University)
Title: Quantum walks in view of discrete geometric analysis
[ pdf ]
This lecture treats quantum walks in view of “geometry of unitary operators”. Ideas in discrete geometric analysis, developed to solve various problems in analysis on graphs, are effectively employed to give new insight to the theory which has been attracting computer scientists and physicists over the past decade.

The notion of quantum walks is a “quantum” version of classical random walks, which is, as indicated by its name, closely related to quantum mechanics. Unlike the case of classical walks depending on imaginary random control by an external device such as the flip of a coin or the cast of a dice, randomness of quantum walks is directly linked to the probabilistic nature of physical states in the quantum system concerned.

Though a proto-idea of quantum walks is already seen in the celebrated theory of path integrals initiated by R. P. Feynman, its intense study has rather a short history, compared with that of classical walks which has been developed and applied to various disciplines for more than a century. Indeed it is in the pioneer work by Y. Aharonov, L. Davidovich, and N. Zagury that the term “quantum (random) walk” was coined for the first time (1993). Since then, several setups for the theory have been proposed, mainly aiming at the design of fast algorithms by means of quantum computing. In this lecture, I take up the simplest formulation among them, and discuss some fundamental aspects of quantum walks. Especially, I discuss periodic quantum walks on crystal lattices, and establish local limit formulae under a certain condition.

A (discrete-time) quantum walk is simply defined to be a pair (U; E) of a unitary operator U acting in a Hilbert space H and a complete orthonormal basis E = fexj x 2 V g of H, where the set of indices V is supposed to be countable. A connection with quantum mechanics is made by regarding E as the set of pure (eigen) states associated with an observable, and thinking of Unex (n 2 Z) as the time evolution of the state ex. Then the quantity pn(x; y) := jhUnex; eyij2 is the probability that the state ey is observed at time n in this evolutional system, and hence we obtain a family of probabilities (probability mass functions) fpn(x; ¢)g1n=1 on V . This is a quantum analogue of n-step transition probability for classical walks. One of main themes in the study of quantum walks is, as in the classical case, asymptotic behaviors of pn(x; y) as n goes to infinity, which turns out to be quite a bit different from the classical case.

The abstract setup above is entirely within the scope of operator theory in Hilbert spaces, thereby, at the first sight, not seeming to invent any new aspects at all. But a simple geometric view opens up new horizons, which lead us to a rich and varied theory of of quantum walks.

Date: December 15 (Tue), 2009, 17:00-18:00
Place: Room 056, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Toshikazu Sunada (砂田利一) (Meiji University)
Title: Open Problems in Discrete Geometric Analysis
[ pdf ]
Discrete geometric analysis is a hybrid field of several traditional disciplines: graph theory, geometry, theory of discrete groups, and probability. This field concerns solely analysis on graphs, a synonym of "1-dimensional cell complex". In this talk, I shall discuss several open problems related to the discrete Laplacian, a "protagonist" in discrete geometric analysis. Topics dealt with are 1. Ramanujan graphs, 2. Spectra of covering graphs, 3. Zeta functions of finitely generated groups.
Date: December 22 (Tue), 2009, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Kyo Nishiyama (西山 享) (Aoyama Gakuin University)
Title: 既約表現の隨伴多様体は余次元 1 で連結か?—証明の破綻とその背景
[ pdf ]
既約 Harish-Chandra $ (g, K) $ 加群の原始イデアルの隨伴多様体が既約であって、ただ一つの冪零隨伴軌道 $ O^G $ の閉包になることはよく知られている(Joseph, Borho)。 一方、HC加群の隨伴多様体は必ずしも既約でないが、その既約成分は $ O^G $ の $ K $-等質ラグランジュ部分多様体の閉包になる。 それらの既約成分は余次元1で連結であることをいくつかの集会で報告したが、その証明には初等的な誤りがあった。セミナーでは、証明の元になった Vogan の定理の紹介 (もちろん間違っていない) と、それを拡張する際になぜ証明が破綻するかについてお話しする。(今のところ証明修復の目処は立っていない。)

© Toshiyuki Kobayashi