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Lie Groups and Representation Theory
Seminar information archive ～05/23｜Next seminar｜Future seminars 05/24～
| Date, time & place | Tuesday 16:30 - 18:00 126Room #126 (Graduate School of Math. Sci. Bldg.) |
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Seminar information archive
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
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
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
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 ]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.
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
http://akagi.ms.u-tokyo.ac.jp/seminar.html
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 ]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.
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
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.
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
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2007.html#20070529neeb
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 ]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.
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
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2007.html#20070525boumuki
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 ]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.
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
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2007.html#20070525kaneyuki
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 ]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 $M$.
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2007.html#20070525kaneyuki
2007/05/22
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
甲斐千舟 (九州大学)
A characterization of symmetric cones by an order-reversing property of the pseudoinverse maps
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
甲斐千舟 (九州大学)
A characterization of symmetric cones by an order-reversing property of the pseudoinverse maps
[ Abstract ]
When a regular open convex cone is given, a natural partial order is introduced into the ambient vector space. If we consider the cone of positive numbers, this partial order is the usual one, and is reversed by taking inverse numbers in the cone. In general, for every symmetric cone, the inverse map of the associated Jordan algebra reverses the order.
In this talk, we investigate this order-reversing property in the class of homogeneous convex cones which is much wider than that of symmetric cones. We show that a homogeneous convex cone is a symmetric cone if and only if the order is reversed by the Vinberg's *-map, which generalizes analytically the inverse maps of Jordan algebras associated with symmetric cones. Actually, our main theorem is formulated in terms of the family of pseudoinverse maps including the Vinberg's *-map as a special one. While our result is a characterization of symmetric cones, also we would like to mention O. Güler's result that for every homogeneous convex cone, some analogous pseudoinverse maps always reverse the order.
[ Reference URL ]When a regular open convex cone is given, a natural partial order is introduced into the ambient vector space. If we consider the cone of positive numbers, this partial order is the usual one, and is reversed by taking inverse numbers in the cone. In general, for every symmetric cone, the inverse map of the associated Jordan algebra reverses the order.
In this talk, we investigate this order-reversing property in the class of homogeneous convex cones which is much wider than that of symmetric cones. We show that a homogeneous convex cone is a symmetric cone if and only if the order is reversed by the Vinberg's *-map, which generalizes analytically the inverse maps of Jordan algebras associated with symmetric cones. Actually, our main theorem is formulated in terms of the family of pseudoinverse maps including the Vinberg's *-map as a special one. While our result is a characterization of symmetric cones, also we would like to mention O. Güler's result that for every homogeneous convex cone, some analogous pseudoinverse maps always reverse the order.
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
2007/05/17
15:00-16:30 Room #002 (Graduate School of Math. Sci. Bldg.)
真野元 (東京大学数理科学研究科)
The unitary inversion operator for the minimal representation of the indefinite orthogonal group O(p,q)
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
真野元 (東京大学数理科学研究科)
The unitary inversion operator for the minimal representation of the indefinite orthogonal group O(p,q)
[ Abstract ]
The indefinite orthogonal group $O(p,q)$ ($p+q$ even, greater than four) has a distinguished infinite dimensional irreducible unitary representation called the 'minimal representation'. Among various models, the $L^2$-model of the minimal representation of $O(p,q)$ was established by Kobayashi-Ørsted (2003). In this talk, we focus on and present an explicit formula for the unitary inversion operator, which plays a key role for the analysis on this L2-model as well as understanding the $G$-action on $L^2(C)$. Our proof uses the Radon transform of distributions supported on the light cone.
This is a joint work with T. Kobayashi.
[ Reference URL ]The indefinite orthogonal group $O(p,q)$ ($p+q$ even, greater than four) has a distinguished infinite dimensional irreducible unitary representation called the 'minimal representation'. Among various models, the $L^2$-model of the minimal representation of $O(p,q)$ was established by Kobayashi-Ørsted (2003). In this talk, we focus on and present an explicit formula for the unitary inversion operator, which plays a key role for the analysis on this L2-model as well as understanding the $G$-action on $L^2(C)$. Our proof uses the Radon transform of distributions supported on the light cone.
This is a joint work with T. Kobayashi.
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
2007/05/08
17:00-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
荒川知幸 (奈良女子大学)
Affine W-algebras and their representations
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
荒川知幸 (奈良女子大学)
Affine W-algebras and their representations
[ Abstract ]
The W-algebras are an interesting class of vertex algebras, which can be understood as a generalization of Virasoro algebra. It was originally introduced by Zamolodchikov in his study of conformal field theory. Later Feigin-Frenkel discovered that the W-algebras can be defined via the method of quantum BRST reduction. A few years ago this method was generalized by Kac-Roan-Wakimoto in full generality, producing many interesting vertex algebras. Almost at the same time Premet re-discovered the finite-dimensional version of W-algebras (finite W-algebras), in connection with the modular representation theory.
In the talk we quickly recall the Feigin-Frenkel theory which connects the Whittaker models of the center of $U({\\mathfrak g})$ and affine (principal) W-algebras, and discuss their representation theory. Next we recall the construction of Kac-Roan-Wakimoto and discuss the representation theory of affine W-algebras associated with general nilpotent orbits. In particular, I explain how the representation theory of finite W-algebras (=the endmorphism ring of the generalized Gelfand-Graev representation) applies to the representation of affine W-algebras.
[ Reference URL ]The W-algebras are an interesting class of vertex algebras, which can be understood as a generalization of Virasoro algebra. It was originally introduced by Zamolodchikov in his study of conformal field theory. Later Feigin-Frenkel discovered that the W-algebras can be defined via the method of quantum BRST reduction. A few years ago this method was generalized by Kac-Roan-Wakimoto in full generality, producing many interesting vertex algebras. Almost at the same time Premet re-discovered the finite-dimensional version of W-algebras (finite W-algebras), in connection with the modular representation theory.
In the talk we quickly recall the Feigin-Frenkel theory which connects the Whittaker models of the center of $U({\\mathfrak g})$ and affine (principal) W-algebras, and discuss their representation theory. Next we recall the construction of Kac-Roan-Wakimoto and discuss the representation theory of affine W-algebras associated with general nilpotent orbits. In particular, I explain how the representation theory of finite W-algebras (=the endmorphism ring of the generalized Gelfand-Graev representation) applies to the representation of affine W-algebras.
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
2007/05/01
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
飯田正敏 (城西大学)
Harish-Chandra expansion of the matrix coefficients of $P_J$ Principal series Representation of $Sp(2,R)$
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
飯田正敏 (城西大学)
Harish-Chandra expansion of the matrix coefficients of $P_J$ Principal series Representation of $Sp(2,R)$
[ Abstract ]
Asymptotic expansion of the matrix coefficents of class-1 principal series representation was considered by Harish-Chandra. The coefficient of the leading exponent of the expansion is called the c-function which plays an important role in the harmonic analysis on the Lie group.
In this talk, we consider the Harish-Chandra expansion of the matrix coefficients of the standard representation which is the parabolic induction with respect to a non-minimal parabolic subgroup of $Sp(2,R)$.
This is the joint study with Professor T. Oda.
[ Reference URL ]Asymptotic expansion of the matrix coefficents of class-1 principal series representation was considered by Harish-Chandra. The coefficient of the leading exponent of the expansion is called the c-function which plays an important role in the harmonic analysis on the Lie group.
In this talk, we consider the Harish-Chandra expansion of the matrix coefficients of the standard representation which is the parabolic induction with respect to a non-minimal parabolic subgroup of $Sp(2,R)$.
This is the joint study with Professor T. Oda.
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html
2007/04/24
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Taro Yoshino (吉野太郎) (University of Tokyo)
Existence problem of compact Clifford-Klein forms of the infinitesimal homogeneous space of indefinite Stiefle manifolds
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2007.html#20070424yoshino
Taro Yoshino (吉野太郎) (University of Tokyo)
Existence problem of compact Clifford-Klein forms of the infinitesimal homogeneous space of indefinite Stiefle manifolds
[ Abstract ]
The existence problem of compact Clifford-Klein forms is important in the study of discrete groups. There are several open problems on it, even in the reductive cases, which is most studied. For a homogeneous space of reductive type, one can define its `infinitesimal' homogeneous space.
This homogeneous space is easier to consider the existence problem of compact Clifford-Klein forms.
In this talk, we especially consider the infinitesimal homogeneous spaces of indefinite Stiefel manifolds. And, we reduce the existence problem of compact Clifford-Klein forms to certain algebraic problem, which was already studied from other motivation.
[ Reference URL ]The existence problem of compact Clifford-Klein forms is important in the study of discrete groups. There are several open problems on it, even in the reductive cases, which is most studied. For a homogeneous space of reductive type, one can define its `infinitesimal' homogeneous space.
This homogeneous space is easier to consider the existence problem of compact Clifford-Klein forms.
In this talk, we especially consider the infinitesimal homogeneous spaces of indefinite Stiefel manifolds. And, we reduce the existence problem of compact Clifford-Klein forms to certain algebraic problem, which was already studied from other motivation.
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2007.html#20070424yoshino
2006/10/31
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
大島利雄氏 (東大数理)
ルート系の部分系の分類
http://akagi.ms.u-tokyo.ac.jp/seminar.html
大島利雄氏 (東大数理)
ルート系の部分系の分類
[ Abstract ]
ルート系 Ξ からルート系 Σ へのCartan整数を保つ写像の分類を考える(像は Σ の部分系と見なせる).
Σ のWeyl群(Σ の内部同型)で移りあうものを同値とみたときの分類をまず行い,
同値の条件をさらに Ξ の自己同型(部分系の分類に対応),Ξ の既約成分の自己同型の直積,
Σ の自己同型などを許すものに広げた場合の分類や像が放物型かどうかの判定も与える.
ルート系の dual pair の概念を定義し,同値類への Ξ の自己同型の作用の考察に用いる.
[ Reference URL ]ルート系 Ξ からルート系 Σ へのCartan整数を保つ写像の分類を考える(像は Σ の部分系と見なせる).
Σ のWeyl群(Σ の内部同型)で移りあうものを同値とみたときの分類をまず行い,
同値の条件をさらに Ξ の自己同型(部分系の分類に対応),Ξ の既約成分の自己同型の直積,
Σ の自己同型などを許すものに広げた場合の分類や像が放物型かどうかの判定も与える.
ルート系の dual pair の概念を定義し,同値類への Ξ の自己同型の作用の考察に用いる.
http://akagi.ms.u-tokyo.ac.jp/seminar.html
2006/07/31
15:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Guster Olafsson (Louisiana State University) 15:00-16:00
The Heat equation, the Segal-Bargmann transform and generalizations - II
[ Reference URL ]
http://akagi.ms.u-tokyo.ac.jp/seminar.html
Boris Rubin (Louisiana State University) 16:30-17:30
Radon transforms on Grassmannians and Matrix Spaces
http://akagi.ms.u-tokyo.ac.jp/seminar.html
Guster Olafsson (Louisiana State University) 15:00-16:00
The Heat equation, the Segal-Bargmann transform and generalizations - II
[ Reference URL ]
http://akagi.ms.u-tokyo.ac.jp/seminar.html
Boris Rubin (Louisiana State University) 16:30-17:30
Radon transforms on Grassmannians and Matrix Spaces
[ Abstract ]
Diverse geometric problems in $R^N$ get a new flavor if a generic point $x=(x_1,...,x_N)$ is regarded as a matrix with appropriately organized entries (set, e.g., $x=(x_{i,j})_{n \\times m}$ for $N=nm$). This well known observation has led to a series of breakthrough achievements in mathematics. In integral geometry it suggests a number of the so-called ``higher-rank" problems when such traditional scalar notions as ``distance", ``angle", and ``scaling" become matrix-valued. I will be speaking about Radon transforms on Grassmann manifolds and matrix spaces and some related problems of harmonic analysis where these phenomena come into play.
[ Reference URL ]Diverse geometric problems in $R^N$ get a new flavor if a generic point $x=(x_1,...,x_N)$ is regarded as a matrix with appropriately organized entries (set, e.g., $x=(x_{i,j})_{n \\times m}$ for $N=nm$). This well known observation has led to a series of breakthrough achievements in mathematics. In integral geometry it suggests a number of the so-called ``higher-rank" problems when such traditional scalar notions as ``distance", ``angle", and ``scaling" become matrix-valued. I will be speaking about Radon transforms on Grassmann manifolds and matrix spaces and some related problems of harmonic analysis where these phenomena come into play.
http://akagi.ms.u-tokyo.ac.jp/seminar.html
2006/07/25
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Boris Rubin (Louisiana State University)
Radon Transforms: Basic Concepts
http://akagi.ms.u-tokyo.ac.jp/seminar.html
Boris Rubin (Louisiana State University)
Radon Transforms: Basic Concepts
[ Abstract ]
How can we reconstruct a function on a manifold from integrals of this function over certain submanifolds?
This is one of the central problems in integral geometry and tomography, which leads to the notion of the Radon transform.
The first talk is of introductory character.
We discuss basic ideas of the original Radon's paper (1917), then proceed to the Minkowski-Funk transform and more general totally geodesic Radon transforms on the $n$-dimensional unit sphere.
The main emphasis is an intimate connection of these transforms with the relevant harmonic analysis.
We will see that Radon transforms of this type and their inverses can be regarded as members of analytic families of suitable convolution operators and successfully studied in the framework of these families.
I also plan to discuss an open problem of small divisors on the unit sphere, which arises in studying injectivity of generalized Minkowski-Funk transforms for non-central spherical sections.
[ Reference URL ]How can we reconstruct a function on a manifold from integrals of this function over certain submanifolds?
This is one of the central problems in integral geometry and tomography, which leads to the notion of the Radon transform.
The first talk is of introductory character.
We discuss basic ideas of the original Radon's paper (1917), then proceed to the Minkowski-Funk transform and more general totally geodesic Radon transforms on the $n$-dimensional unit sphere.
The main emphasis is an intimate connection of these transforms with the relevant harmonic analysis.
We will see that Radon transforms of this type and their inverses can be regarded as members of analytic families of suitable convolution operators and successfully studied in the framework of these families.
I also plan to discuss an open problem of small divisors on the unit sphere, which arises in studying injectivity of generalized Minkowski-Funk transforms for non-central spherical sections.
http://akagi.ms.u-tokyo.ac.jp/seminar.html
2006/07/20
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Guster Olafsson (Louisiana State University)
The Heat equation, the Segal-Bargmann transform and generalizations - I
[ Reference URL ]
http://akagi.ms.u-tokyo.ac.jp/seminar.html
Guster Olafsson (Louisiana State University)
The Heat equation, the Segal-Bargmann transform and generalizations - I
[ Reference URL ]
http://akagi.ms.u-tokyo.ac.jp/seminar.html
2006/07/11
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
縫田 光司 (産業技術総合研究所)
On the isomorphism problem of Coxeter groups and related topics
http://akagi.ms.u-tokyo.ac.jp/seminar.html
縫田 光司 (産業技術総合研究所)
On the isomorphism problem of Coxeter groups and related topics
[ Abstract ]
コクセター群について、そのコクセター系としての同型類がコクセターグラフと一対一対応することは周知の事実であるが、一方で抽象群としての同型類は(ワイル群の場合に限っても)そのような対応をしていない。今回は、この同型類の決定問題について、その歴史のあらまし(特に、無限群も含めた一般の場合について、10年ほど前まで殆ど何の結果も得られていなかったことは特筆に価する)と、近年の研究の進展状況を、具体例や関連する結果を交えつつ紹介する。
[ Reference URL ]コクセター群について、そのコクセター系としての同型類がコクセターグラフと一対一対応することは周知の事実であるが、一方で抽象群としての同型類は(ワイル群の場合に限っても)そのような対応をしていない。今回は、この同型類の決定問題について、その歴史のあらまし(特に、無限群も含めた一般の場合について、10年ほど前まで殆ど何の結果も得られていなかったことは特筆に価する)と、近年の研究の進展状況を、具体例や関連する結果を交えつつ紹介する。
http://akagi.ms.u-tokyo.ac.jp/seminar.html
2006/06/13
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
織田 寛 (拓殖大学工学部)
古典型複素Lie環の一般Verma加群に対する最小多項式
織田 寛 (拓殖大学工学部)
古典型複素Lie環の一般Verma加群に対する最小多項式
[ Abstract ]
古典型複素Lie環 g の自然表現から自然に定まる U(g) 係数の正方行列を F とする.g のスカラー一般Verma加群 $M_Θ(λ)$ に対して,複素モニック多項式 q(x) で q(F) の各成分が全て Ann $M_Θ(λ)$ に属するような最小次数のものを “$M_Θ(λ)$ の最小多項式” とよぶ.M(λ) を $M_Θ(λ)$ を商加群とするVerma加群とし,q(F) の各成分と Ann M(λ) が生成する U(g) の両側イデアルを $I_Θ(λ)$ とすると,最近
(1) 各λに対する $M_Θ(λ)$ の最小多項式の明示公式
(2) $M_Θ(λ)= M(λ)/I_Θ(λ)M(λ)$ が成り立つためのλの 必要十分条件
が得られた(これらは大島により g = gln の場合には既に得られている).セミナーでは(2)を示すための q(F) の各成分の Harish-Chandra 準同型像の計算法を主に説明する.
古典型複素Lie環 g の自然表現から自然に定まる U(g) 係数の正方行列を F とする.g のスカラー一般Verma加群 $M_Θ(λ)$ に対して,複素モニック多項式 q(x) で q(F) の各成分が全て Ann $M_Θ(λ)$ に属するような最小次数のものを “$M_Θ(λ)$ の最小多項式” とよぶ.M(λ) を $M_Θ(λ)$ を商加群とするVerma加群とし,q(F) の各成分と Ann M(λ) が生成する U(g) の両側イデアルを $I_Θ(λ)$ とすると,最近
(1) 各λに対する $M_Θ(λ)$ の最小多項式の明示公式
(2) $M_Θ(λ)= M(λ)/I_Θ(λ)M(λ)$ が成り立つためのλの 必要十分条件
が得られた(これらは大島により g = gln の場合には既に得られている).セミナーでは(2)を示すための q(F) の各成分の Harish-Chandra 準同型像の計算法を主に説明する.
2006/05/16
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
大島 利雄 (東京大学大学院数理科学研究科)
確定特異点型の可換な微分作用素系について
大島 利雄 (東京大学大学院数理科学研究科)
確定特異点型の可換な微分作用素系について
[ Abstract ]
実簡約Lie群やその対称空間をコンパクト多様体に実現すると,不変微分作用素系はその境界に沿って確定特異点を持つ可換微分作用素系となる.
可換微分作用素系がただ一つの作用素から特徴づけられることを基に,境界の近傍で多価解析的な同時固有関数の一般的構成を考察し,表現論,特にWhittaker模型などへの応用を論じる(Harish-Chandra同型やGoodman-Wallach作用素の微分方程式論の立場からの解釈などを含む).
実簡約Lie群やその対称空間をコンパクト多様体に実現すると,不変微分作用素系はその境界に沿って確定特異点を持つ可換微分作用素系となる.
可換微分作用素系がただ一つの作用素から特徴づけられることを基に,境界の近傍で多価解析的な同時固有関数の一般的構成を考察し,表現論,特にWhittaker模型などへの応用を論じる(Harish-Chandra同型やGoodman-Wallach作用素の微分方程式論の立場からの解釈などを含む).
2006/04/18
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
伴 克馬 (東京大学大学院数理科学研究科)
Rankin-Cohen-伊吹山型の微分作用素について
伴 克馬 (東京大学大学院数理科学研究科)
Rankin-Cohen-伊吹山型の微分作用素について
[ Abstract ]
正則保型形式の正則微分は一般に保型形式ではなくなるが、それらを組み合わせることで新たな正則保型形式を与えることもできる。
楕円モジュラー形式に対するRankin-Cohen微分作用素はその最も簡単な例である。伊吹山はSiegelモジュラー形式に対するこのようなタイプの微分作用素がどのような形をしているかについて、一般的な記述を与えた。
今回のセミナーでは、伊吹山による結果を表現論的な命題として捉え直し、その命題が自然にSU(p,q)やO*(2p)上の正則保型形式にも拡張されることを説明する。
正則保型形式の正則微分は一般に保型形式ではなくなるが、それらを組み合わせることで新たな正則保型形式を与えることもできる。
楕円モジュラー形式に対するRankin-Cohen微分作用素はその最も簡単な例である。伊吹山はSiegelモジュラー形式に対するこのようなタイプの微分作用素がどのような形をしているかについて、一般的な記述を与えた。
今回のセミナーでは、伊吹山による結果を表現論的な命題として捉え直し、その命題が自然にSU(p,q)やO*(2p)上の正則保型形式にも拡張されることを説明する。
