Lie Groups and Representation Theory

Seminar information archive ~03/28Next seminarFuture seminars 03/29~

Date, time & place Tuesday 16:30 - 18:00 126Room #126 (Graduate School of Math. Sci. Bldg.)

Seminar information archive

2008/07/08

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
直井 克之 (東大数理)
construction of extended affine Lie algebras from multiloop Lie algebras
[ Abstract ]
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。
[ Reference URL ]
http://akagi.ms.u-tokyo.ac.jp/seminar.html

2008/07/01

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
奥田 隆幸 (東大数理)
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 ]
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。
TypeIV 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 (mod6) 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。

2008/06/03

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
示野 信一 (岡山理科大)
Matrix valued commuting differential operators with B2 symmetry
[ Abstract ]
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。
[ Reference URL ]
http://akagi.ms.u-tokyo.ac.jp/seminar.html

2008/05/27

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
笹木集夢 (早稲田大学)
Visible actions on multiplicity-free spaces
[ Abstract ]
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.
[ Reference URL ]
http://akagi.ms.u-tokyo.ac.jp/seminar.html

2008/05/20

16:45-18:15   Room #126 (Graduate School of Math. Sci. Bldg.)
吉野太郎 (東京工業大学)
Lipsman予想の反例と代数多様体の特異点について
[ Abstract ]
リー群$G$が多様体$M$に作用しているとき, その商空間$G\\backspace M$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.
[ Reference URL ]
http://akagi.ms.u-tokyo.ac.jp/seminar.html

2008/05/13

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
加藤晃史 (東京大学)
On endomorphisms of the Weyl algebra
[ Abstract ]
Noncommutative geometry has revived the interest in the Weyl algebras, which are basic building blocks of quantum field theories.
The Weyl algebra $A_n(\\C)$ is an associative algebra over $\\C$ generated by $p_i, q_i$ ($i=1,\\cdots,n$) with relations $[p_i, q_j]=\\delta_{ij}$. Every endomorphism of $A_n$ is injective since $A_n$ is simple.
Dixmier (1968) initiated a systematic study of the Weyl algebra $A_1$ and posed the following problem: Is every endomorphism of $A_1$ an automorphism?
We give an affirmative answer to this conjecture.
[ Reference URL ]
http://akagi.ms.u-tokyo.ac.jp/seminar.html

2008/01/22

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
大島 利雄 (東京大学)
Connecion problems for Fuchsian differential equations free from accessory parameters
[ Abstract ]
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.
[ Reference URL ]
http://akagi.ms.u-tokyo.ac.jp/seminar.html

2008/01/17

17:00-18:00   Room #123 (Graduate School of Math. Sci. Bldg.)
手塚勝貴 (東大数理)
Proper actions of SL(2,R) on irreducible complex symmetric spaces
[ Abstract ]
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.
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

2008/01/15

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Fulton Gonzalez (Tufts University)
Group contractions, invariant differential operators and the matrix Radon transform


[ Abstract ]
Let $M_{n,k}$ denote the vector space of real $n\\times k$ matrices.
The matrix motion group is the semidirect product $(\\text O(n)\\times \\text O(k))\\ltimes M_{n,k}$, and is the Cartan motion group
associated with the real Grassmannian $G_{n,n+k}$.
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.
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

2007/12/18

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
阿部 紀行 (東京大学)
On the existence of homomorphisms between principal series of complex
semisimple Lie groups
[ Abstract ]
The principal series representations of a semisimple Lie group play an important role in the representation theory. We study the principal series representation of a complex semisimple Lie group and determine when there exists a nonzero homomorphism between the representations.
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

2007/12/11

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
井上順子 (鳥取大学)
Characterization of some smooth vectors for irreducible representations of exponential solvable Lie groups
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

2007/11/20

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
西山 享 (京都大学)
Asymptotic cone for semisimple elements and the associated variety of degenerate principal series
[ Abstract ]
Let $ a $ be a hyperbolic element in a semisimple Lie algebra over the real number field. Let $ K $ be the complexification of a maximal compact subgroup of the corresponding real adjoint group. We study the asymptotic cone of the semisimple orbit through $ a $ under the adjoint action by $ K $. The resulting asymptotic cone is the associated variety of a degenerate principal series representation induced from the parabolic associated to $ a $.
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

2007/11/06

15:00-16:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Michaël Pevzner (Université de Reims and University of Tokyo)
Quantization of symmetric spaces and representation. IV
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

2007/11/06

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
森脇政泰 (広島大学)
Multiplicity-free decompositions of the minimal representation of the indefinite orthogonal group
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

2007/11/01

16:30-18:00   Room #052 (Graduate School of Math. Sci. Bldg.)
Michaël Pevzner (Université de Reims and University of Tokyo)
Quantization of symmetric spaces and representation. III
[ Abstract ]
Kontsevich's formality theorem and applications in Representation theory.

We shall first give an explicit construction of an associative star-product on an arbitrary smooth finite-dimensional Poisson manifold.

As application, we will consider in details the crucial example of the dual of a finite-dimensional Lie algebra and will sketch a generalization of the Duflo isomorphism describing the set of infinitesimal characters of irreducible unitary representations of the corresponding Lie group.
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

2007/10/30

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
松本久義 (東京大学大学院数理科学研究科)
On Weyl groups for parabolic subalgebras
[ Abstract ]
Let ${\\mathfrak g}$ be a complex semisimple Lie algebra.
We call a parabolic subalgebra ${\\mathfrak p}$ of ${\\mathfrak g}$
normal, if any parabolic subalgebra which has a common Levi part with ${\\mathfrak p}$
is conjugate to ${\\mathfrak p}$ under an inner automorphism of ${\\mathfrak g}$.
For a normal parabolic subalgebra, we have a good notion of the restricted root system
or the little Weyl group. We have a comparison result on the Bruhat order on the Weyl group for
${\\mathfrak g}$ and the little Weyl group.
We also apply this result to the existence problem of the homomorphisms between scalar generalized Verma modules.
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

2007/10/30

15:00-16:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Michaël Pevzner (Université de Reims and University of Tokyo)
Quantization of symmetric spaces and representation. II
[ Abstract ]
Back to Mathematics. Two methods of quantization.

We will start with a discussion on

-Weyl symbolic calculus on a symplectic vector space
and its asymptotic behavior.


In the second part, as a consequence of previous considerations, we will define the notion of deformation quantization.
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

2007/10/25

16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Michael Pevzner (Universite de Reims and University of Tokyo)
Quantization of symmetric spaces and representations. I
[ Abstract ]
The first and introductory lecture of a series of four will be devoted to the discussion of fundamental principles of the Quantum mechanics and their mathematical formulation. This part is not essential for the rest of the course but it might give a global vision of the subject to be considered.

We shall introduce the Weyl symbolic calculus, that relates classical and quantum observables, and will explain its relationship with the so-called deformation quantization of symplectic manifolds.

Afterwards, we will pay attention to a more algebraic question of formal deformation of an arbitrary smooth Poisson manifold and will define the Kontsevich star-product.
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

2007/10/09

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Michael Pevzner (Reims University and University of Tokyo)
Rankin-Cohen brackets and covariant quantization
[ Abstract ]
The particular geometric structure of causal symmetric spaces permits the definition of a covariant quantization of these homogeneous manifolds.
Composition formulae (#-products) of quantizad operators give rise to a new interpretation of Rankin-Cohen brackets and allow to connect them with the branching laws of tensor products of holomorphic discrete series representations.
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

2007/10/02

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Pablo Ramacher (Gottingen University)


Invariant integral operators on affine G-varieties and their kernels
[ Abstract ]
We consider certain invariant integral operators on a smooth affine variety M carrying the action of a reductive algebraic group G, and assume that G acts on M with an open orbit. Then M is isomorphic to a homogeneous vector bundle, and can locally be described via the theory of prehomogenous vector spaces. We then study the Schwartz kernels of the considered operators, and give a description of their singularities using the calculus of b-pseudodifferential operators developed by Melrose. In particular, the restrictions of the kernels to the diagonal can be described in terms of local zeta functions.
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

2007/06/29

15:30-17:45   Room #122 (Graduate School of Math. Sci. Bldg.)
Salem Ben Said (Nancy大)
On the theory of Bessel functions associated with root systems
[ Abstract ]
The theory of spherical functions on Riemannian symmetric spaces G/K and on non-compactly causal symmetric spaces G/H has a long and rich history. In this talk we will show how one can use a limit transition approach to obtain generalized Bessel functions on flat symmetric spaces via the spherical functions. A similar approach can be applied to the theory of Heckman-Opdam hypergeometric functions to investigate generalized Bessel functions related to arbitrary root system. We conclude the talk by giving a conjecture about the nature and order of the singularities of the Bessel functions related to non-compactly causal symmetric spaces.
[ Reference URL ]
http://akagi.ms.u-tokyo.ac.jp/seminar.html

2007/06/19

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
原岡喜重氏 (熊本大学)
Rigid local systemとその切断の積分表示,および接続係数
[ Abstract ]
A local system on $CP^1-\\{finite points\\}$ is called physically rigid if it is uniquely determined up to isomorphisms by the local monodromies. We explain two algorithms to construct every physically rigid local systems. By applying the algorithms we obtain integral representations of solutions of the corresponding Fuchsian differential equation. Moreover we can express connection coefficients of the equation in terms of the integrals. These results will be applied to several differential equations arising from the representation theory.

2007/05/29

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Karl-Hermann Neeb (Technische Universität Darmstadt)
A host algebra for the regular representations of the canonical commutation relations
[ Abstract ]
We report on joint work with H. Grundling (Sydney).
The concept of a host algebra generalises that of a group $C^*$-algebra to groups which are not locally compact in the sense that its non-degenerate representations are in one-to-one correspondence with representations of the group under consideration. A full host algebra covering all continuous unitary representations exist for an abelian topological group if and only if it (essentially) has a locally compact completion. Therefore one has to content oneselves with certain classes of representations covered by a host algebra. We show that there exists a host algebra for the set of continuous representations of the countably dimensional Heisenberg group corresponding to a non-zero central character.
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2007.html#20070529neeb

2007/05/25

14:30-16:00   Room #122 (Graduate School of Math. Sci. Bldg.)
坊向伸隆 (大阪市立大学)
The classification of simple irreducible pseudo-Hermitian symmetric spaces: from a view of elliptic orbits
[ Abstract ]
In this talk, we call a special elliptic element an Spr-element, we create an equivalence relation on the set of Spr-elements of a real form of a complex simple Lie algebra, and we classify Spr-elements of each real form of all complex simple Lie algebras under our equivalence relation. Besides, we demonstrate that the classification of Spr-elements under our equivalence relation corresponds to that of simple irreducible pseudo-Hermitian symmetric Lie algebras under Berger's equivalence relation. In terms of the correspondence, we achieve the classification of simple irreducible pseudo-Hermitian symmetric Lie algebras without Berger's classification.
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2007.html#20070525boumuki

2007/05/25

16:00-17:30   Room #122 (Graduate School of Math. Sci. Bldg.)
金行壮二 (上智大学名誉教授)
Causalities, G-structures and symmetric spaces
[ Abstract ]
Let M be an $n$-dimensional smooth manifold, $F(M)$ the frame bundle of $M$, and let $G$ be a Lie subgroup of $GL(n,\\mathbb R)$. We say that $M$ has a $G$-structure, if there exists a principal subbundle $Q$ of $F(M)$ with structure group $G$. Let $C$ be a causal cone in $\\mathbb R^n$, and let $Aut C$ denote the automorphism group of $C$.

Starting from a causal structure $\\mathcal{C}$ on $M$ with model cone $C$, we construct an $Aut C$-structure $Q(\\mathcal{C})$. Several concepts on causal structures can be interpreted as those on $Aut C$-structures. As an example, the causal automorphism group $Aut(M,\\mathcal{C})$ of $M$ coincides with the automorphism group $Aut(M,Q(\\mathcal{C}))$ of the $Aut C$-structure.

As an application, we will consider the unique extension of a local causal transformation on a Cayley type symmetric space $M$ to the global causal automorphism of the compactification of $M$.
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2007.html#20070525kaneyuki

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