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Lie Groups and Representation Theory
Seminar information archive ～05/01｜Next seminar｜Future seminars 05/02～
| Date, time & place | Tuesday 16:30 - 18:00 126Room #126 (Graduate School of Math. Sci. Bldg.) |
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Seminar information archive
2008/10/28
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Joachim Hilgert (Paderborn University)
Chevalley's restriction theorem for supersymmetric Riemannian symmetric spaces
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
Joachim Hilgert (Paderborn University)
Chevalley's restriction theorem for supersymmetric Riemannian symmetric spaces
[ Abstract ]
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)
[ Reference URL ]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)
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
2008/10/21
17:00-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
落合啓之 (名古屋大学)
Invitation to Atlas combinatorics
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
落合啓之 (名古屋大学)
Invitation to Atlas combinatorics
[ Abstract ]
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。
[ Reference URL ]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。
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
2008/10/14
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Jan Moellers (Paderborn University)
The Dirichlet-to-Neumann map as a pseudodifferential
operator
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
Jan Moellers (Paderborn University)
The Dirichlet-to-Neumann map as a pseudodifferential
operator
[ Abstract ]
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 $\\partial M$ the Dirichlet-to-Neumann operator $\\Lambda_g$ maps Dirichlet boundary data $f$ to the corresponding Neumann boundary data $\\Lambda_g f =(\\partial_\\nu u)|_{\\partial M}$ where $u$ denotes the unique solution to the Dirichlet problem $\\laplace_g u=0$ in $M$, $u|_{\\partial M} = f$.
The main statement is that this operator is a first order elliptic pseudodifferential operator on the boundary $\\partial M$.
We will first give a brief overview of how to define the Dirichlet-to-Neumann operator as a map $\\Lambda_g:H^{1/2}(\\partial M)\\longrightarrow H^{-1/2}(\\partial M)$ 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.
[ Reference URL ]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 $\\partial M$ the Dirichlet-to-Neumann operator $\\Lambda_g$ maps Dirichlet boundary data $f$ to the corresponding Neumann boundary data $\\Lambda_g f =(\\partial_\\nu u)|_{\\partial M}$ where $u$ denotes the unique solution to the Dirichlet problem $\\laplace_g u=0$ in $M$, $u|_{\\partial M} = f$.
The main statement is that this operator is a first order elliptic pseudodifferential operator on the boundary $\\partial M$.
We will first give a brief overview of how to define the Dirichlet-to-Neumann operator as a map $\\Lambda_g:H^{1/2}(\\partial M)\\longrightarrow H^{-1/2}(\\partial M)$ 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.
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
2008/09/08
11:00-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Federico Incitti (ローマ第 1 大学)
Dyck partitions, quasi-minuscule quotients and Kazhdan-Lusztig polynomials
http://akagi.ms.u-tokyo.ac.jp/seminar.html
Federico Incitti (ローマ第 1 大学)
Dyck partitions, quasi-minuscule quotients and Kazhdan-Lusztig polynomials
[ Abstract ]
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.
[ Reference URL ]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.
http://akagi.ms.u-tokyo.ac.jp/seminar.html
2008/07/29
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
小木曽 岳義 (城西大学)
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)
小木曽 岳義 (城西大学)
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 ]
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。
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。
2008/07/15
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
廣惠 一希 (東大数理)
GL(4,R)の退化主系列表現の一般Whittaker関数
http://akagi.ms.u-tokyo.ac.jp/seminar.html
廣惠 一希 (東大数理)
GL(4,R)の退化主系列表現の一般Whittaker関数
[ Abstract ]
$GL(n,R)$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$C^\\infty$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.
[ Reference URL ]$GL(n,R)$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$C^\\infty$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.
http://akagi.ms.u-tokyo.ac.jp/seminar.html
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
http://akagi.ms.u-tokyo.ac.jp/seminar.html
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 ]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。
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
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。
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
http://akagi.ms.u-tokyo.ac.jp/seminar.html
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 ]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。
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
http://akagi.ms.u-tokyo.ac.jp/seminar.html
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 ]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.
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
http://akagi.ms.u-tokyo.ac.jp/seminar.html
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 ]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.
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
http://akagi.ms.u-tokyo.ac.jp/seminar.html
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 ]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.
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
http://akagi.ms.u-tokyo.ac.jp/seminar.html
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 ]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.
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
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
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 ]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.
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
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
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 ]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.
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
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
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 ]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.
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
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
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
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 ]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 $.
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
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
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
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
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 ]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.
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
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
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 ]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.
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
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
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 ]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.
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
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
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 ]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.
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
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
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 ]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.
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
