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Lie Groups and Representation Theory
Seminar information archive ～04/24｜Next seminar｜Future seminars 04/25～
| Date, time & place | Tuesday 16:30 - 18:00 126Room #126 (Graduate School of Math. Sci. Bldg.) |
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Seminar information archive
2010/04/15
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Uuganbayar Zunderiya (Nagoya University)
Generalized hypergeometric systems (ENGLISH)
Uuganbayar Zunderiya (Nagoya University)
Generalized hypergeometric systems (ENGLISH)
[ Abstract ]
A new type of hypergeometric differential equations was introduced and studied by H. Sekiguchi. The investigated system of partial differential equation generalizes the Gauss-Aomoto-Gelfand system which in its turn stems from the classical set of differential relations for the solutions to a generic algebraic equation introduced by K.Mayr in 1937. Gauss-Aomoto-Gelfand systems can be expressed as the determinants of $2\\times 2$ matrices of derivations with respect to certain variables. H. Sekiguchi generalized this construction by looking at determinations of arbitrary $l\\times l$ matrices of derivations with respect to certain variables.
In this talk we study the dimension of global (and local) solutions to the generalized systems of Gauss-Aomoto-Gelfand hypergeometric systems. The main results in the talk are a combinatorial formula for the dimension of global (and local) solutions of the generalized Gauss-Aomoto-Gelfand system and a theorem on generic holonomicity of a certain class of such systems.
A new type of hypergeometric differential equations was introduced and studied by H. Sekiguchi. The investigated system of partial differential equation generalizes the Gauss-Aomoto-Gelfand system which in its turn stems from the classical set of differential relations for the solutions to a generic algebraic equation introduced by K.Mayr in 1937. Gauss-Aomoto-Gelfand systems can be expressed as the determinants of $2\\times 2$ matrices of derivations with respect to certain variables. H. Sekiguchi generalized this construction by looking at determinations of arbitrary $l\\times l$ matrices of derivations with respect to certain variables.
In this talk we study the dimension of global (and local) solutions to the generalized systems of Gauss-Aomoto-Gelfand hypergeometric systems. The main results in the talk are a combinatorial formula for the dimension of global (and local) solutions of the generalized Gauss-Aomoto-Gelfand system and a theorem on generic holonomicity of a certain class of such systems.
2010/04/06
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
加藤周 (京都大学)
On the characters of tempered modules of affine Hecke
algebras of classical type
加藤周 (京都大学)
On the characters of tempered modules of affine Hecke
algebras of classical type
[ Abstract ]
We present an inductive algorithm to compute the characters
of tempered modules of an affine Hecke algebras of classical
types, based on a new class of representations which we call
"tempered delimits". They have some geometric origin in the
eDL correspondence.
Our new algorithm has some advantage to the Lusztig-Shoji
algorithm (which also describes the characters of tempered
modules via generalized Green functions) in the sense it
enables us to tell how the characters of tempered modules
changes as the parameters vary.
This is a joint work with Dan Ciubotaru at Utah.
We present an inductive algorithm to compute the characters
of tempered modules of an affine Hecke algebras of classical
types, based on a new class of representations which we call
"tempered delimits". They have some geometric origin in the
eDL correspondence.
Our new algorithm has some advantage to the Lusztig-Shoji
algorithm (which also describes the characters of tempered
modules via generalized Green functions) in the sense it
enables us to tell how the characters of tempered modules
changes as the parameters vary.
This is a joint work with Dan Ciubotaru at Utah.
2010/02/19
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Yves Benoist (Orsay)
Discrete groups acting on homogeneous spaces V
Yves Benoist (Orsay)
Discrete groups acting on homogeneous spaces V
[ Abstract ]
I will focus on recent advances on our understanding of discrete subgroups of Lie groups.
I will first survey how ideas from semisimple algebraic groups, ergodic theory and representation theory help us to understand properties of these discrete subgroups.
I will then focus on a joint work with Jean-Francois Quint studying the dynamics of these discrete subgroups on finite volume homogeneous spaces and proving the following result:
We fix two integral matrices A and B of size d, of determinant 1, and such that no finite union of vector subspaces is invariant by A and B. We fix also an irrational point on the d-dimensional torus. We will then prove that for n large the set of images of this point by the words in A and B of length at most n becomes equidistributed in the torus.
I will focus on recent advances on our understanding of discrete subgroups of Lie groups.
I will first survey how ideas from semisimple algebraic groups, ergodic theory and representation theory help us to understand properties of these discrete subgroups.
I will then focus on a joint work with Jean-Francois Quint studying the dynamics of these discrete subgroups on finite volume homogeneous spaces and proving the following result:
We fix two integral matrices A and B of size d, of determinant 1, and such that no finite union of vector subspaces is invariant by A and B. We fix also an irrational point on the d-dimensional torus. We will then prove that for n large the set of images of this point by the words in A and B of length at most n becomes equidistributed in the torus.
2010/02/02
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Fanny Kassel (Orsay)
Deformation of compact quotients of homogeneous spaces
Fanny Kassel (Orsay)
Deformation of compact quotients of homogeneous spaces
[ Abstract ]
Let G/H be a reductive homogeneous space. In all known examples, if
G/H admits compact Clifford-Klein forms, then it admits "standard"
ones, by uniform lattices of some reductive subgroup L of G acting
properly on G/H. In order to obtain more generic Clifford-Klein forms,
we prove that for L of real rank 1, if one slightly deforms in G a
uniform lattice of L, then its action on G/H remains properly
discontinuous. As an application, we obtain compact quotients of SO(2,2n)/U(1,n) by Zariski-dense discrete subgroups of SO(2,2n) acting properly discontinuously.
Let G/H be a reductive homogeneous space. In all known examples, if
G/H admits compact Clifford-Klein forms, then it admits "standard"
ones, by uniform lattices of some reductive subgroup L of G acting
properly on G/H. In order to obtain more generic Clifford-Klein forms,
we prove that for L of real rank 1, if one slightly deforms in G a
uniform lattice of L, then its action on G/H remains properly
discontinuous. As an application, we obtain compact quotients of SO(2,2n)/U(1,n) by Zariski-dense discrete subgroups of SO(2,2n) acting properly discontinuously.
2010/01/12
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
西岡斉治 (東京大学大学院数理科学研究科博士課程)
代数的差分方程式の可解性と既約性
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20100112nishioka
西岡斉治 (東京大学大学院数理科学研究科博士課程)
代数的差分方程式の可解性と既約性
[ Abstract ]
差分代数の理論を使って,代数的差分方程式の代数函数解や超幾
何函数解の非存在や,存在する場合の特殊解の分類をする。
[ Reference URL ]差分代数の理論を使って,代数的差分方程式の代数函数解や超幾
何函数解の非存在や,存在する場合の特殊解の分類をする。
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20100112nishioka
2009/12/22
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
西山享 (青山学院大学)
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
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
西山享 (青山学院大学)
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
[ Abstract ]
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。)
[ Reference URL ]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。)
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
2009/12/15
17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
砂田利一氏 (明治大学理工学部)
Open Problems in Discrete Geometric Analysis
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2009.html#20091215sunada
砂田利一氏 (明治大学理工学部)
Open Problems in Discrete Geometric Analysis
[ Abstract ]
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.
[ Reference URL ]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.
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2009.html#20091215sunada
2009/11/04
16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Gert Heckman (IMAPP, Faculty of Science, Radboud University Nijmegen)
Birational Hyperbolic Geometry
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
Gert Heckman (IMAPP, Faculty of Science, Radboud University Nijmegen)
Birational Hyperbolic Geometry
[ Abstract ]
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.
[ Reference URL ]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.
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
2009/10/15
16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)
土岡俊介 (RIMS, Kyoto University)
Hecke-Clifford superalgebras and crystals of type $D^{(2)}_{l}$
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
土岡俊介 (RIMS, Kyoto University)
Hecke-Clifford superalgebras and crystals of type $D^{(2)}_{l}$
[ Abstract ]
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 $D^{(2)}_{l}$.
[ Reference URL ]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 $D^{(2)}_{l}$.
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
2009/10/13
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
小寺諒介 (東京大学)
Extensions between finite-dimensional simple modules over a generalized current Lie algebra
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
小寺諒介 (東京大学)
Extensions between finite-dimensional simple modules over a generalized current Lie algebra
[ Abstract ]
$\\mathfrak{g}$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.
[ Reference URL ]$\\mathfrak{g}$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.
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
2009/08/12
10:00-16:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Sigurdur Helgason (MIT) 10:00-11:00
Radon Transform and some Applications
Fulton G. Gonzalez (Tufts University) 11:20-12:20
Multitemporal Wave Equations: Mean Value Solutins
Angela Pasquale (Universite Metz) 14:00-15:00
Analytic continuation of the resolvent of the Laplacian in the Euclidean settings
Decay of smooth vectors for regular representations
Sigurdur Helgason (MIT) 10:00-11:00
Radon Transform and some Applications
Fulton G. Gonzalez (Tufts University) 11:20-12:20
Multitemporal Wave Equations: Mean Value Solutins
Angela Pasquale (Universite Metz) 14:00-15:00
Analytic continuation of the resolvent of the Laplacian in the Euclidean settings
[ 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.
Henrik Schlichtkrull (University of Copenhagen) 15:30-16:30We 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.
Decay of smooth vectors for regular representations
[ 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. Krotz)
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. Krotz)
2009/06/15
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Vladimir P. Kostov (Nice大学)
On the Schur-Szeg\\"o composition of polynomials
Vladimir P. Kostov (Nice大学)
On the Schur-Szeg\\"o composition of polynomials
[ Abstract ]
The Schur-Szeg\\"o composition of the degree $n$ polynomials $P:=\\sum_{j=0}^na_jx^j$ and $Q:=\\sum_{j=0}^nb_jx^j$ is defined by the formula $P*Q:=\\sum_{j=0}^na_jb_jx^j/C_n^j$ where $C_n^j=n!/j!(n-j)!$. Every degree $n$ polynomial having one of its roots at $-1$ (i.e. $P=(x+1)(x^{n-1}+c_1x^{n-2}+\\cdots +c_{n-1})$) is representable as a Schur-Szeg\\"o composition of $n-1$ polynomials of the form $(x+1)^{n-1}(x+a_i)$ where the numbers $a_i$ are uniquely defined up to permutation. Denote the elementary symmetric polynomials of the numbers $a_i$ by $\\sigma_1$, $\\ldots$, $\\sigma_{n-1}$. The talk will focus on some properties of the affine mapping
$$(c_1,\\ldots ,c_{n-1})\\mapsto (\\sigma_1,\\ldots ,\\sigma_{n-1})$$
The Schur-Szeg\\"o composition of the degree $n$ polynomials $P:=\\sum_{j=0}^na_jx^j$ and $Q:=\\sum_{j=0}^nb_jx^j$ is defined by the formula $P*Q:=\\sum_{j=0}^na_jb_jx^j/C_n^j$ where $C_n^j=n!/j!(n-j)!$. Every degree $n$ polynomial having one of its roots at $-1$ (i.e. $P=(x+1)(x^{n-1}+c_1x^{n-2}+\\cdots +c_{n-1})$) is representable as a Schur-Szeg\\"o composition of $n-1$ polynomials of the form $(x+1)^{n-1}(x+a_i)$ where the numbers $a_i$ are uniquely defined up to permutation. Denote the elementary symmetric polynomials of the numbers $a_i$ by $\\sigma_1$, $\\ldots$, $\\sigma_{n-1}$. The talk will focus on some properties of the affine mapping
$$(c_1,\\ldots ,c_{n-1})\\mapsto (\\sigma_1,\\ldots ,\\sigma_{n-1})$$
2009/02/03
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Gombodorj Bayarmagnai (東京大学数理科学研究科)
The (g,K)-module structure of principal series and related Whittaker functions of SU(2,2)
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
Gombodorj Bayarmagnai (東京大学数理科学研究科)
The (g,K)-module structure of principal series and related Whittaker functions of SU(2,2)
[ Abstract ]
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 $(\\mathfrak g,K)$-module structure on that type of principal series explicitely, and provide various integral expressions of some smooth Whittaker functions with certain $K$-types.
[ Reference URL ]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 $(\\mathfrak g,K)$-module structure on that type of principal series explicitely, and provide various integral expressions of some smooth Whittaker functions with certain $K$-types.
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
2009/01/15
13:30-17:20 Room #050 (Graduate School of Math. Sci. Bldg.)
柏原正樹 (京都大学数理解析研究所) 13:30-14:30
Quantization of complex manifolds
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/index.files/oshima60th200901.html
小林俊行 (東京大学大学院数理科学研究科) 15:00-16:00
Global geometry on locally symmetric spaces — beyond the Riemannian case
Classification of Fuchsian systems and their connection problem
柏原正樹 (京都大学数理解析研究所) 13:30-14:30
Quantization of complex manifolds
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/index.files/oshima60th200901.html
小林俊行 (東京大学大学院数理科学研究科) 15:00-16:00
Global geometry on locally symmetric spaces — beyond the Riemannian case
[ Abstract ]
The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry.
In contrast, in areas such as Lorentz geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry, as well as in various other kinds of geometry (symplectic, complex geometry, ...), surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure.
In this talk, I plan to give an exposition on the recent developments on the question about the global natures of locally non-Riemannian homogeneous spaces, with emphasis on the existence problem of compact forms, rigidity and deformation.
大島利雄 (東京大学大学院数理科学研究科) 16:20-17:20The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry.
In contrast, in areas such as Lorentz geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry, as well as in various other kinds of geometry (symplectic, complex geometry, ...), surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure.
In this talk, I plan to give an exposition on the recent developments on the question about the global natures of locally non-Riemannian homogeneous spaces, with emphasis on the existence problem of compact forms, rigidity and deformation.
Classification of Fuchsian systems and their connection problem
[ Abstract ]
We explain a classification of Fuchsian systems on the Riemann sphere together with Katz's middle convolution, Yokoyama's extension and their relation to a Kac-Moody root system discovered by Crawley-Boevey.
Then we present a beautifully unified connection formula for the solution of the Fuchsian ordinary differential equation without moduli and apply the formula to the harmonic analysis on a symmetric space.
We explain a classification of Fuchsian systems on the Riemann sphere together with Katz's middle convolution, Yokoyama's extension and their relation to a Kac-Moody root system discovered by Crawley-Boevey.
Then we present a beautifully unified connection formula for the solution of the Fuchsian ordinary differential equation without moduli and apply the formula to the harmonic analysis on a symmetric space.
2009/01/12
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
西岡斉治 (東京大学大学院数理科学研究科博士課程)
代数的差分方程式の可解性と既約性
西岡斉治 (東京大学大学院数理科学研究科博士課程)
代数的差分方程式の可解性と既約性
[ Abstract ]
差分代数の理論を使って,代数的差分方程式の代数函数解や超幾
何函数解の非存在や,存在する場合の特殊解の分類をする。
差分代数の理論を使って,代数的差分方程式の代数函数解や超幾
何函数解の非存在や,存在する場合の特殊解の分類をする。
2008/12/04
17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Genkai Zhang (Chalmers and Gothenburg University)
Realization of quanternionic discrete series as spaces of H-holomorphic
functions
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
Genkai Zhang (Chalmers and Gothenburg University)
Realization of quanternionic discrete series as spaces of H-holomorphic
functions
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
2008/12/02
17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
金井雅彦 (名古屋大学)
消滅と剛性
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
金井雅彦 (名古屋大学)
消滅と剛性
[ Abstract ]
The aim of my talk is to reveal an unforeseen link between the classical vanishing theorems of Matsushima and Weil, on the one hand, and rigidity 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.
[ Reference URL ]The aim of my talk is to reveal an unforeseen link between the classical vanishing theorems of Matsushima and Weil, on the one hand, and rigidity 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.
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
2008/11/25
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
吉野太郎 (東工大)
$\\mathbb R^n$への$\\mathbb R^2$の固有な作用と周期性
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
吉野太郎 (東工大)
$\\mathbb R^n$への$\\mathbb R^2$の固有な作用と周期性
[ Abstract ]
Consider $\\R^2$ actions on $\\R^n$ 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 $n\\geq 5$. And, we also find a periodicity on such counterexamples.
As a key lemma, we use "proper analogue" of the five lemma on
exact sequence.
[ Reference URL ]Consider $\\R^2$ actions on $\\R^n$ 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 $n\\geq 5$. And, we also find a periodicity on such counterexamples.
As a key lemma, we use "proper analogue" of the five lemma on
exact sequence.
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
2008/11/18
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Jorge Vargas (FAMAF-CIEM, C\'ordoba)
Liouville measures and multiplicity formulae for admissible restriction of Discrete Series
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
Jorge Vargas (FAMAF-CIEM, C\'ordoba)
Liouville measures and multiplicity formulae for admissible restriction of Discrete Series
[ Abstract ]
Let $H \\subset G$ be reductive matrix Lie groups. We fix a square integrable irreducible representation $\\pi$ of $G.$
Let $\\Omega $ denote the coadjoint orbit of the Harish-Chandra parameter of $\\pi.$
Assume $\\pi$ 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 $\\pi$ restricted to $H$ and push forward to $\\mathfrak h^\\star$ of the Liouville measure for $\\Omega.$ This generalizes work of Duflo-Heckman-Vergne.
[ Reference URL ]Let $H \\subset G$ be reductive matrix Lie groups. We fix a square integrable irreducible representation $\\pi$ of $G.$
Let $\\Omega $ denote the coadjoint orbit of the Harish-Chandra parameter of $\\pi.$
Assume $\\pi$ 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 $\\pi$ restricted to $H$ and push forward to $\\mathfrak h^\\star$ of the Liouville measure for $\\Omega.$ This generalizes work of Duflo-Heckman-Vergne.
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
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
