## Lie Groups and Representation Theory

Seminar information archive ～03/20｜Next seminar｜Future seminars 03/21～

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

### 2008/05/13

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

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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