## Lie群論・表現論セミナー

開催情報 火曜日　16:30～18:00　数理科学研究科棟(駒場) 126号室 小林俊行 https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2008年05月13日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室

On endomorphisms of the Weyl algebra
[ 講演概要 ]
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.
[ 参考URL ]
http://akagi.ms.u-tokyo.ac.jp/seminar.html

### 2008年01月22日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室

Connecion problems for Fuchsian differential equations free from accessory parameters
[ 講演概要 ]
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.
[ 参考URL ]
http://akagi.ms.u-tokyo.ac.jp/seminar.html

### 2008年01月17日(木)

17:00-18:00   数理科学研究科棟(駒場) 123号室

Proper actions of SL(2,R) on irreducible complex symmetric spaces
[ 講演概要 ]
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.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2008年01月15日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室
Fulton Gonzalez 氏 (Tufts University)
Group contractions, invariant differential operators and the matrix Radon transform

[ 講演概要 ]
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.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007年12月18日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室

On the existence of homomorphisms between principal series of complex
semisimple Lie groups
[ 講演概要 ]
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.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007年12月11日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室

Characterization of some smooth vectors for irreducible representations of exponential solvable Lie groups
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007年11月20日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室

Asymptotic cone for semisimple elements and the associated variety of degenerate principal series
[ 講演概要 ]
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 $a$.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007年11月06日(火)

15:00-16:30   数理科学研究科棟(駒場) 126号室
Michaël Pevzner 氏 (Université de Reims and University of Tokyo)
Quantization of symmetric spaces and representation. IV
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007年11月06日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室

Multiplicity-free decompositions of the minimal representation of the indefinite orthogonal group
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007年11月01日(木)

16:30-18:00   数理科学研究科棟(駒場) 052号室
Michaël Pevzner 氏 (Université de Reims and University of Tokyo)
Quantization of symmetric spaces and representation. III
[ 講演概要 ]
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.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007年10月30日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室

On Weyl groups for parabolic subalgebras
[ 講演概要 ]
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.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007年10月30日(火)

15:00-16:30   数理科学研究科棟(駒場) 126号室
Michaël Pevzner 氏 (Université de Reims and University of Tokyo)
Quantization of symmetric spaces and representation. II
[ 講演概要 ]
Back to Mathematics. Two methods of quantization.

-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.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007年10月25日(木)

16:30-18:00   数理科学研究科棟(駒場) 002号室

Michael Pevzner 氏 (Universite de Reims and University of Tokyo)
Quantization of symmetric spaces and representations. I
[ 講演概要 ]
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.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007年10月09日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室
Michael Pevzner 氏 (Reims University and University of Tokyo)
Rankin-Cohen brackets and covariant quantization
[ 講演概要 ]
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.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007年10月02日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室
Pablo Ramacher 氏 (Gottingen University)

Invariant integral operators on affine G-varieties and their kernels
[ 講演概要 ]
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.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007年06月29日(金)

15:30-17:45   数理科学研究科棟(駒場) 122号室

Salem Ben Said 氏 (Nancy大)
On the theory of Bessel functions associated with root systems
[ 講演概要 ]
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.
[ 参考URL ]
http://akagi.ms.u-tokyo.ac.jp/seminar.html

### 2007年06月19日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室

Rigid local systemとその切断の積分表示,および接続係数
[ 講演概要 ]
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年05月29日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室

Karl-Hermann Neeb 氏 (Technische Universität Darmstadt)
A host algebra for the regular representations of the canonical commutation relations
[ 講演概要 ]
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.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2007.html#20070529neeb

### 2007年05月25日(金)

14:30-16:00   数理科学研究科棟(駒場) 122号室
いつもと曜日・時刻が違います

The classification of simple irreducible pseudo-Hermitian symmetric spaces: from a view of elliptic orbits
[ 講演概要 ]
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.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2007.html#20070525boumuki

### 2007年05月25日(金)

16:00-17:30   数理科学研究科棟(駒場) 122号室
いつもと曜日時刻がちがいますのでご注意ください

Causalities, G-structures and symmetric spaces
[ 講演概要 ]
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 $M$.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2007.html#20070525kaneyuki

### 2007年05月22日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室

A characterization of symmetric cones by an order-reversing property of the pseudoinverse maps
[ 講演概要 ]
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.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007年05月17日(木)

15:00-16:30   数理科学研究科棟(駒場) 002号室
いつもと、曜日・時刻・場所が違いますのでご注意ください

The unitary inversion operator for the minimal representation of the indefinite orthogonal group O(p,q)
[ 講演概要 ]
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.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007年05月08日(火)

17:00-18:00   数理科学研究科棟(駒場) 126号室
この週には同氏による集中講義 14:40-16:40 が行われます。セミナーの時刻はいつもと違いますのでご注意ください。

Affine W-algebras and their representations
[ 講演概要 ]
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.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007年05月01日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室

Harish-Chandra expansion of the matrix coefficients of $P_J$ Principal series Representation of $Sp(2,R)$
[ 講演概要 ]
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.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html

### 2007年04月24日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室
Taro Yoshino (吉野太郎) 氏 (University of Tokyo)
Existence problem of compact Clifford-Klein forms of the infinitesimal homogeneous space of indefinite Stiefle manifolds

[ 講演概要 ]
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.
[ 参考URL ]
https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2007.html#20070424yoshino