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Lie Groups and Representation Theory Seminar 2008

List of speakers:
Fulton Gonzalez, Katsuki Teduka, Toshio Oshima, Akishi Kato, Taro Yoshino (1), Atsumu Sasaki, Nobukazu Shimeno, Takayuki Okuda, Katsuyuki Naoi, Kazuki Hiroe, Takeyoshi Kogiso, Federico Incitti, Joachim Hilgert (1), Jan Moellers, Hiroyuki Ochiai, Joachim Hilgert (2), Jorge Vargas, Taro Yoshino (2), Masahiko Kanai, Genkai Zhang
Date: January 15 (Tue), 2008, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Fulton Gonzalez (Tufts University)
Title: Group contractions, invariant differential operators and the matrix Radon transform
Abstract:
[ pdf ]
Let Mn,k denote the vector space of real n × k matrices. The matrix motion group is the semidirect product (O(n) × O(k)) \ltimes Mn,k , and is the Cartan motion group associated with the real Grassmannian Gn,n+k . The matrix Radon transform is an integral transform associated with a double fibration involving homogeneous spaces of this group. We provide a set of algebraically independent generators of the subalgebra of its universal enveloping algebra invariant under the Adjoint representation. One of the elements of this set characterizes the range of the matrix Radon transform.
Date: January 17 (Thu), 2008, 17:00-18:00
Place: Room 123, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Katsuki Teduka (手塚勝貴) (University of Tokyo)
Title: Proper actions of SL(2,R) on irreducible complex symmetric spaces
Abstract:
[ pdf ]
We determine the irreducible complex symmetric spaces on which SL(2,R) acts properly. We use the T. Kobayashi's criterion for the proper actions. Also we use the symmetry or unsymmetry of the weighted Dynkin diagram of the theory of nilpotent orbits.
Date: January 22 (Tue), 2008, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Toshio Oshima (大島利雄) (University of Tokyo)
Title: Connection problems for Fuchsian differential equations free from accessory parameters
Abstract:
[ pdf ]
The classification of Fuchsian equations without accessory parameters was formulated as Deligne-Simpson problem, which was solved by Katz and they are studied by Haraoka and Yokoyama. If the number of singular points of such equations is three, they have no geometric moduli. We give a unified connection formula for such differential equations as a conjecture and show that it is true for the equations whose local monodromy at a singular point has distinct eigenvalues. Other Fuchsian differential equations with accessory parameters and hypergeometric functions with multi-variables are also discussed.
Date: May 13 (Tue), 2008, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Akishi Kato (加藤晃史) (University of Tokyo)
Title: On endomorphisms of the Weyl algebra
Abstract:
[ pdf ]
Noncommutative geometry has revived the interest in the Weyl algebras, which are basic building blocks of quantum field theories. The Weyl algebra An(C) is an associative algebra over C generated by pi, qi (i = 1, ..., n) with relations [pi, qj] = δij. Every endomorphism of An is injective since An is simple. Dixmier (1968) initiated a systematic study of the Weyl algebra A1 and posed the following problem: Is every endomorphism of A1 an automorphism? We give an affirmative answer to this conjecture.
Date: May 20 (Tue), 2008, 16:45-18:15
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Taro Yoshino (吉野太郎) (Tokyo Institute of Technology)
Title: Lipsman予想の反例と代数多様体の特異点について
Abstract:
[ pdf ]
リー群 G が多様体 M に作用しているとき, その商空間 G\M のハウスドルフ性は, 不連続群論の研究において重要である. 特に, ベキ零リー群が線型空間にアファインかつ自由に作用するとき, 商位相は常にハウスドルフであるとLipsmanは予想した. しかし, この予想には反例があり, 商位相は必ずしもハウスドルフでない. この講演では, この非ハウスドルフ性を '可視化' したい. より正確には, M への G 作用から, 自然に代数多様体 V が定義され, V の特異点が商位相の非ハウスドルフ性に対応することを見る.
Date: May 27 (Tue), 2008, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Atsumu Sasaki (笹木集夢) (Waseda University)
Title: Visible actions on multiplicity-free spaces
Abstract:
[ pdf ]

The holomorphic action of a Lie group G on a complex manifold D is called strongly visible if there exist a real submanifold S such that D' := G • S is open in D and an anti-holomorphic diffeomorphism σ which is an identity map on S and preserves each G-orbit in D'.

In this talk, we treat the case where D is a multiplicity-free space V of a connected complex reductive Lie group G(C), and show that the action of a compact real form of G(C) on V is strongly visible.

Date: June 3 (Tue), 2008, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Nobukazu Shimeno (示野信一) (Okayama University of Science)
Title: Matrix valued commuting differential operators with B2 symmetry
Abstract:
[ pdf ]
B2 型のWeyl群の作用による対称性を持つ2次正方行列値の2階の可換な微分作用素を構成した。 作用素は Iida (Publ. Res. Inst. Math. Sci. Kyoto Univ. 32 (1996)) により計算された Sp(2,R)/U(2) の等質ベクトル束上の不変微分作用素の動径成分を特別な場合として含み、係数は楕円関数を用いて表される。 講演では、群の場合、可換な作用素の構成、spin Calogero-Sutherland 模型との関係について述べる。
Date: July 1 (Tue), 2008, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Takayuki Okuda (奥田隆幸) (University of Tokyo)
Title: 不変式の zeta 多項式の零点と,微分作用素の関係について
Abstract:
[ pdf ]
MacWilliams 変換と呼ばれる変換で不変な複素 2 変数斉次多項式に対して,zeta 多項式と呼ばれる複素 1 変数多項式を定義する. Type IV extremal と呼ばれる不変式の無限列に対し,deg = 0 (mod 6) の場合には,対応する全ての zeta 多項式の零点が同一円周上に乗るという事が証明されているが,deg = 2,4 (mod 6) の場合は未解決であった. この講演では,不変式に対する微分作用素を用いて,deg = 4 (mod 6) の場合にも全ての zeta 多項式の零点が同一円周上に乗るということを示したい.
Date: July 8 (Tue), 2008, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Katsuyuki Naoi (直井克之) (University of Tokyo)
Title: Construction of extended affine Lie algebras from multiloop Lie algebras
Abstract:
[ pdf ]
affine Lie algebra の Kac-Moody Lie algebra とは異なる一般化として,extended affine Lie algebra と呼ばれる Lie algebra の class を考える. ほとんどの extended affine Lie algebra は,有限次元 simple Lie algebra と,有限個の互いに可換な有限位数自己同型を用いて構成できることがすでに知られている.
この講演では,上の構成によって得られる extended affine Lie algebra がどのような場合に(適当な意味で)同型となるか,という問題に関する結果をお話ししたい.
Date: July 15 (Tue), 2008, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Kazuki Hiroe (廣惠一希) (University of Tokyo)
Title: GL(4,R) の退化主系列表現の一般 Whittaker 関数
Abstract:
[ pdf ]
GL(n,R) の退化球主系列表現の一般 Whittaker 模型の空間は,対称空間 GL(n,R)/O(n) 上の C 級関数の中で,ある微分作用素達の kernel として特徴付けられる.この微分作用素達は,大島利雄氏による退化主系列表現に対する Poisson 変換の像の特徴付けに用いられたものであり,その明示的な表示が氏によって得られている.また,こうした kernel 定理は山下博氏のユニタリ最低ウエイト加群の一般 Whittaker 模型に対する定理の類似にあたる.こういった背景の下,GL(4,R) の退化主系列表現に対し,いくつかの具体例を考えたい.そこでは一般 Whittaker 模型は一変数変形 Bessel 関数,Horn の二変数合流型超幾何関数によって実現される.
Date: July 29 (Tue), 2008, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Takeyoshi Kogiso (小木曽岳義) (Josai University)
Title: Clifford 代数の表現から作られる局所関数等式を満たす多項式とそれに付随する空間について(佐藤文広氏との共同研究)
Abstract:
[ pdf ]
概均質ベクトル空間の理論の基本定理(局所関数等式)は,大雑把に言うと,正則概均質ベクトル空間の相対不変式の複素ベキの Fourier 変換が双対概均質ベクトル空間の相対不変式の複素ベキにガンマ因子をかけたものと一致することを主張している.
この講演では,概均質ベクトル空間の相対不変式ではないにもかかわらず,その複素ベキが同種の局所関数等式を満たすような多項式が,Clifford 代数の表現より構成できることを報告する.
Date: September 8 (Mon), 2008, 11:00-12:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Federico Incitti (Sapienza Università di Roma)
Title: Dyck partitions, quasi-minuscule quotients and Kazhdan-Lusztig polynomials
Abstract:
[ pdf ]
Kazhdan-Lusztig polynomials were first defined by Kazhdan and Lusztig in [Invent. Math., 53 (1979), 165-184]. Since then, numerous applications have been found, especially to representation theory and to the geometry of Schubert varieties. In 1987 Deodhar introduced parabolic analogues of these polynomials. These are related to their ordinary counterparts in several ways, and also play a direct role in other areas, including geometry of partial flag manifolds and the theory of Macdonald polynomials.

In this talk I study the parabolic Kazhdan-Lusztig polynomials of the quasi-minuscule quotients of the symmetric group. More precisely, I will first show how these quotients are closely related to ''rooted partitions'' and then I will give explicit, closed combinatorial formulas for the polynomials. These are based on a special class of rooted partitions the ''rooted-Dyck'' partitions, and imply that they are always (either zero or) a power of q.

I will conclude with some enumerative results on Dyck and rooted-Dyck partitions, showing a connection with random walks on regular trees.

This is partly based on a joint work with Francesco Brenti and Mario Marietti.

(GCOE Lectures)
Date and place:
  1. October 14 (Tue), 2008, 15:00-16:00, Room 118
  2. October 15 (Wed), 2008, 15:00-16:00, Room 122
  3. October 16 (Thu), 2008, 15:00-16:00, Room 123
  4. October 17 (Fri), 2008, 15:00-16:00, Room 118
  5. October 27 (Mon), 2008, 16:30-17:30, Room 128
Speaker: Joachim Hilgert (Paderborn University)
Title: Holomorphic extensions of unitary representations
Abstract:
[ pdf ]
  1. Overview and Examples
    In this lecture we present the Gelfand-Gindikin program of decomposing L2-spaces into families of irreducible representations using complex geometry. We then briefly outline results due to Olshanski, Hilgert-Ólafsson-Ørsted, Hilgert-Neeb-Ørsted, Krötz-Stanton and others in this direction. In particular, we will explain holomorphic extensions of holomorphic discrete series representations and their relation to Hardy and weighted Bergman spaces.
  2. Geometric Background
    In this lecture we will explain the complex geometry needed to understand the phenomena described in the first lecture. The key words here are Olshanski semigroups, invariant cones in Lie algebras, Akhiezer-Gindikin domain, and coadjoint orbits of convex type.
  3. Highest weight representations
    In this lecture we explain the extension results in a little more detail and explain how they lead to geometric realizations of singular highest weight representations on nilpotent coadjoint orbits.
  4. Applications and open problems
    In this lecture we present further applications of the given extension results and describe some open problems. In particular, we will mention estimates for automorphic forms (Krötz-Stanton), random matrices (Huckleberry-Püttmann-Zirnbauer), unitarizability of highest weight representation with non-scalar lowest K-type, and infinite dimensional groups.
  5. Holomorphic extensions of highest weight representations to Olshanskii semigroups
    In this lecture I will present a proof of Olshanskii's Theorem, which says that for a simple group of Hermitean type unitarizable highest weight representations can be holomorphically extended to contractive representations of a complex semigroup containing the group in its boundary.
Date: October 14 (Tue), 2008, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Jan Moellers (Paderborn University)
Title: The Dirichlet-to-Neumann map as a pseudodifferential operator
Abstract:
[ pdf ]
Both Dirichlet and Neumann boundary conditions for the Laplace equation are of fundamental importance in Mathematics and Physics. Given a compact connected Riemannian manifold M with boundary ∂M the Dirichlet-to-Neumann operator Λg maps Dirichlet boundary data f to the corresponding Neumann boundary data Λg f = (∂ν u)|M where u denotes the unique solution to the Dirichlet problem Δg u = 0 in M, u|M = f. The main statement is that this operator is a first order elliptic pseudodifferential operator on the boundary ∂M.

We will first give a brief overview of how to define the Dirichlet-to-Neumann operator as a map Λg: H1/2(∂M) → H-1/2(∂M) between Sobolev spaces. In order to show that it is actually a pseudodifferential operator we introduce tangential pseudodifferential operators. This allows us to derive a microlocal factorization of the Laplacian near boundary points. Together with a regularity statement for the heat equation this will finally give the main result.

Date: October 21 (Tue), 2008, 17:00-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Hiroyuki Ochiai (落合啓之) (Nagoya University)
Title: Invitation to Atlas combinatorics
Abstract:
[ pdf ]
この講演は、今月上旬に京大数理研で行なったものと同じです。

半単純リー群のユニタリ表現の分類を手がける Atlas project (J. Adams, D. Vogan らが主催) では、実簡約 (real reductive) 線形代数群の admissible 表現をパラメトライズし、それに関するいくつかのプログラムが公開されています。ウェブサイトはwww.liegroups.org。 現在、そのメインとなるものは Kazhdan-Lusztig-Vogan 多項式です。リー群として複素単純リー群を実リー群と見なしたケースが、通常の Kazhdan-Lusztig 理論に一致し、それを、ある一方向に拡張したのがここで扱われる KLV 理論と考えられます。

この講演では、リー群に関する背景説明などは軽く済ませ、Atlas で公開されているプログラムにおける方言、特に入出力の読み方を通常の言葉に言い換えることで、プログラムを使ってもらう入り口での障壁を減らしたいと考えています。 ふむ、なかなか、使えるな、自分もインストールしてみようか、と思ってもらえれば、成功です。

なお、サーベイトークなので私のオリジナルな結果は含まれていません。また、計算機を使ってデモをする予定です。京都では計算機と板書の切り替えでばたばたしたので、照準を絞って慌てないように話したいと思います。

Date: October 28 (Tue), 2008, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Joachim Hilgert (Paderborn University)
Title: Chevalley's restriction theorem for supersymmetric Riemannian symmetric spaces
Abstract:
[ pdf ]
We start by explaining the concept of a supersymmetric Riemannian symmetric spaces and present the examples studied by Zirnbauer in the context of universality classes of random matrices. For these classes we then show how to formulate and prove an analog of Chevalley's restriction theorem for invariant super-functions.

This is joint work with A. Alldridge (Paderborn) and M. Zirnbauer (Cologne).

Date: November 18 (Tue), 2008, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Jorge Vargas (FAMAF-CIEM, Córdoba)
Title: Liouville measures and multiplicity formulae for admissible restriction of Discrete Series
Abstract:
[ pdf ]
Let HG be reductive matrix Lie groups. We fix a square integrable irreducible representation π of G. Let Ω denote the coadjoint orbit of the Harish-Chandra parameter of π.

Assume π restricted to H is admissible. In joint work with Michel Duflo, by means of "discrete" and "continuos" Heaviside functions we relate the multiplicity of each irreducible H-factor of π restricted to H and push forward to \mathfrak{h} of the Liouville measure for Ω. This generalizes work of Duflo-Heckman-Vergne.

Date: November 25 (Tue), 2008, 16:30-18:00
Place: Room 126, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Taro Yoshino (吉野太郎) (Tokyo Institute of Technology)
Title: Proper R2-actions on Rn and their periodicity
Abstract:
[ pdf ]
Consider R2 actions onRn which is free, affine and unipotent. Our concern here is to answer the following question:

"Does the quotient topology admits a manifold structure?"

Under some weak assumption, we classify all actions up to conjugate, and give a complete answer to the question.

If Lipsman's conjecture were true, all of the answer should be affirmative.

But, we shall find a unique action which gives a negative answer for each n ≥ 5. And, we also find a periodicity on such counterexamples.

As a key lemma, we use "proper analogue" of the five lemma on exact sequence.

(Joint seminar with Topology Seminar)
Date: December 2 (Tue), 2008, 17:00-18:00
Place: Room 056, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Masahiko Kanai (金井雅彦) (Nagoya University)
Title: Vanishing and Rigidity
Abstract:
[ pdf ]
The aim of my talk is to reveal an unforeseen link between the classical vanishing theorems of Matsushima and Weil, on the one hand, andrigidity of the Weyl chamber flow, a dynamical system arising from a higher-rank noncompact Lie group, on the other.

The connection is established via "transverse extension theorems": Roughly speaking, they claim that a tangential 1-form of the orbit foliation of the Weyl chamber flow that is tangentially closed (and satisfies a certain mild additional condition) can be extended to a closed 1-form on the whole space in a canonical manner. In particular, infinitesimal rigidity of the orbit foliation of the Weyl chamber flow is proved as an application.

Date: December 4 (Thu), 2008, 17:00-18:00
Place: Room 056, Graduate School of Mathematical Sciences, the University of Tokyo
Speaker: Genkai Zhang (Chalmers Univ. of Tech./Göteborg Univ.)
Title: Realization of quanternionic discrete series as spaces of H-holomorphic functions
Abstract:
[ pdf ]
Gross and Wallach introduced a family of representations on quaternionic symmetric spaces and studied their continuation. I shall give a realization of the series as H-holomophic vector-valued functions defined in terms of the hypercomplex structure and Dirac operators, and find the reproducing kernel expansion on the quaternionic unit ball.
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© Toshiyuki Kobayashi