## Lie群論・表現論セミナー

過去の記録 ～05/28｜次回の予定｜今後の予定 05/29～

開催情報 | 火曜日 16:30～18:00 数理科学研究科棟(駒場) 126号室 |
---|---|

担当者 | 小林俊行 |

セミナーURL | https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar.html |

**過去の記録**

### 2013年12月17日(火)

16:30-17:30 数理科学研究科棟(駒場) 126号室

非コンパクト型対称空間におけるポアソン変換の$L^{2}$-値域の特徴

付け (JAPANESE)

**貝塚公一 氏**(筑波大学大学院 数理物質科学研究科)非コンパクト型対称空間におけるポアソン変換の$L^{2}$-値域の特徴

付け (JAPANESE)

[ 講演概要 ]

Characterizations of the joint eigenspaces of invariant

differential operators in terms of the Poisson transform have been one of the central problems in harmonic analysis on symmetric spaces.

From the point of view of spectral theory, Strichartz (J. Funct.

Anal.(1989)) formulated a conjecture concerning a certain image

characterization of the Poisson transform of the $L^{2}$-space on the boundary on symmetric spaces of noncompact type. In this talk, we employ techniques in scattering theory to present a positive answer to the Strichartz conjecture.

Characterizations of the joint eigenspaces of invariant

differential operators in terms of the Poisson transform have been one of the central problems in harmonic analysis on symmetric spaces.

From the point of view of spectral theory, Strichartz (J. Funct.

Anal.(1989)) formulated a conjecture concerning a certain image

characterization of the Poisson transform of the $L^{2}$-space on the boundary on symmetric spaces of noncompact type. In this talk, we employ techniques in scattering theory to present a positive answer to the Strichartz conjecture.

### 2013年11月19日(火)

16:30-17:30 数理科学研究科棟(駒場) 126号室

Horospheres, wonderfull compactification and c-function (JAPANESE)

**Simon Gindikin 氏**(Rutgers University (USA))Horospheres, wonderfull compactification and c-function (JAPANESE)

[ 講演概要 ]

I will discuss what is closures of horospheres at the wonderfull compactification and how does it connected with horospherical transforms, c-functions and product-formulas

I will discuss what is closures of horospheres at the wonderfull compactification and how does it connected with horospherical transforms, c-functions and product-formulas

### 2013年11月11日(月)

16:30-17:30 数理科学研究科棟(駒場) 126号室

Alternating sign matrices, primed shifted tableaux and Tokuyama

factorisation theorems (ENGLISH)

**Ronald King 氏**(the University of Southampton)Alternating sign matrices, primed shifted tableaux and Tokuyama

factorisation theorems (ENGLISH)

[ 講演概要 ]

Twenty years ago Okada established a remarkable set of identities relating weighted sums over half-turn alternating sign matrices (ASMs) to products taking the form of deformations of Weyl denominator formulae for Lie algebras B_n, C_n and D_n. Shortly afterwards Simpson added another such identity to the list. It will be shown that various classes of ASMs are in bijective correspondence with certain sets of shifted tableaux, and that statistics on these ASMs may be expressed in terms of the entries in corresponding compass point matrices (CPMs). This then enables the Okada and Simpson identities to be expressed in terms of weighted sums over primed shifted tableaux. This offers the possibility of extending each of these identities, that originally involved a single parameter and a single shifted tableau shape, to more general identities involving both sequences of parameters and shapes specified by arbitrary partitions. It is conjectured that in each case an appropriate multi-parameter weighted sum can be expressed as a product of a deformed Weyl denominator and group character of the type first proved in the A_n case by Tokuyma in 1988. The conjectured forms of the generalised Okada and Simpson identities will be given explicitly, along with an account of recent progress made in collaboration with Angèle Hamel in proving some of them.

Twenty years ago Okada established a remarkable set of identities relating weighted sums over half-turn alternating sign matrices (ASMs) to products taking the form of deformations of Weyl denominator formulae for Lie algebras B_n, C_n and D_n. Shortly afterwards Simpson added another such identity to the list. It will be shown that various classes of ASMs are in bijective correspondence with certain sets of shifted tableaux, and that statistics on these ASMs may be expressed in terms of the entries in corresponding compass point matrices (CPMs). This then enables the Okada and Simpson identities to be expressed in terms of weighted sums over primed shifted tableaux. This offers the possibility of extending each of these identities, that originally involved a single parameter and a single shifted tableau shape, to more general identities involving both sequences of parameters and shapes specified by arbitrary partitions. It is conjectured that in each case an appropriate multi-parameter weighted sum can be expressed as a product of a deformed Weyl denominator and group character of the type first proved in the A_n case by Tokuyma in 1988. The conjectured forms of the generalised Okada and Simpson identities will be given explicitly, along with an account of recent progress made in collaboration with Angèle Hamel in proving some of them.

### 2013年11月07日(木)

13:30-14:20 数理科学研究科棟(駒場) 000号室

擬リーマン局所等質空間上の大域幾何と解析 (ENGLISH)

**小林俊行 氏**(東京大学大学院数理科学研究科)擬リーマン局所等質空間上の大域幾何と解析 (ENGLISH)

[ 講演概要 ]

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 of general signature, surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure.

Taking anti-de Sitter manifolds, which are locally modelled on AdS^n as an example, I plan to explain two programs:

1. (global shape) Exisitence problem of compact locally homogeneous spaces, and defomation theory.

2. (spectral analysis) Construction of the spectrum of the Laplacian, and its stability under the deformation of the geometric structure.

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 of general signature, surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure.

Taking anti-de Sitter manifolds, which are locally modelled on AdS^n as an example, I plan to explain two programs:

1. (global shape) Exisitence problem of compact locally homogeneous spaces, and defomation theory.

2. (spectral analysis) Construction of the spectrum of the Laplacian, and its stability under the deformation of the geometric structure.

### 2013年11月07日(木)

14:30-17:40 数理科学研究科棟(駒場) 000号室

Weightless cohomology of algebraic varieties (ENGLISH)

Visible actions on generalized flag varieties

--- Geometry of multiplicity-free representations of $SO(N)$ (ENGLISH)

Holomorphic discrete series and Borel-de Siebenthal discrete series (ENGLISH)

Branching laws and the local Langlands correspondence (ENGLISH)

**Vaibhav Vaish 氏**(the Institute of Mathematical Sciences) 14:30-15:20Weightless cohomology of algebraic varieties (ENGLISH)

[ 講演概要 ]

Using Morel's weight truncations in categories of mixed sheaves, we attach to any variety defined over complex numbers, over finite fields or even over a number field, a series of groups called the weightless cohomology groups. These lie between the usual cohomology and the intersection cohomology, have a natural ring structure, satisfy Kunneth, and are functorial for certain morphisms.

The construction is motivic and naturally arises in the context of Shimura Varieties where they capture the cohomology of Reductive Borel-Serre compactification. The construction also yields invariants of singularities associated with the combinatorics of the boundary divisors in any resolution.

Using Morel's weight truncations in categories of mixed sheaves, we attach to any variety defined over complex numbers, over finite fields or even over a number field, a series of groups called the weightless cohomology groups. These lie between the usual cohomology and the intersection cohomology, have a natural ring structure, satisfy Kunneth, and are functorial for certain morphisms.

The construction is motivic and naturally arises in the context of Shimura Varieties where they capture the cohomology of Reductive Borel-Serre compactification. The construction also yields invariants of singularities associated with the combinatorics of the boundary divisors in any resolution.

**Yuichiro Tanaka 氏**(the University of Tokyo) 15:40-16:10Visible actions on generalized flag varieties

--- Geometry of multiplicity-free representations of $SO(N)$ (ENGLISH)

[ 講演概要 ]

The subject of study is tensor product representations of irreducible representations of the orthogonal group, which are multiplicity-free. Here we say a group representation is multiplicity-free if any irreducible representation occurs at most once in its irreducible decomposition.

The motivation is the theory of visible actions on complex manifolds, which was introduced by T. Kobayashi. In this theory, the main tool for proving the multiplicity-freeness property of group representations is the ``propagation theorem of the multiplicity-freeness property". By using this theorem and Stembridge's classification result, we obtain the following: All the multiplicity-free tensor product representations of $SO(N)$ and $Spin(N)$ can be obtained from character, alternating tensor product and spin representations combined with visible actions on orthogonal generalized flag varieties.

The subject of study is tensor product representations of irreducible representations of the orthogonal group, which are multiplicity-free. Here we say a group representation is multiplicity-free if any irreducible representation occurs at most once in its irreducible decomposition.

The motivation is the theory of visible actions on complex manifolds, which was introduced by T. Kobayashi. In this theory, the main tool for proving the multiplicity-freeness property of group representations is the ``propagation theorem of the multiplicity-freeness property". By using this theorem and Stembridge's classification result, we obtain the following: All the multiplicity-free tensor product representations of $SO(N)$ and $Spin(N)$ can be obtained from character, alternating tensor product and spin representations combined with visible actions on orthogonal generalized flag varieties.

**Pampa Paul 氏**(Indian Statistical Institute, Kolkata) 16:10-16:40Holomorphic discrete series and Borel-de Siebenthal discrete series (ENGLISH)

[ 講演概要 ]

Let $G_0$ be a simply connected non-compact real simple Lie group with maximal compact subgroup $K_0$.

Let $T_0\\subset K_0$ be a Cartan subgroup of $K_0$ as well as of $G_0$. So $G_0$ has discrete series representations.

Denote by $\\frak{g}, \\frak{k},$ and $\\frak{t}$, the

complexifications of the Lie algebras $\\frak{g}_0, \\frak{k}_0$ and $\\frak{t}_0$ of $G_0, K_0$ and $T_0$ respectively.

There exists a positive root system $\\Delta^+$ of $\\frak{g}$ with respect to $\\frak{t}$, known as the Borel-de Siebenthal positive system for which there is exactly one non-compact simple root, denoted $\\nu$. Let $\\mu$ denote the highest root.

If $G_0/K_0$ is Hermitian symmetric, then $\\nu$ has coefficient $1$ in $\\mu$ and one can define holomorphic discrete series representation of $G_0$ using $\\Delta^+$.

If $G_0/K_0$ is not Hermitian symmetric, the coefficient of $\\nu$ in the highest root $\\mu$ is $2$.

In this case, Borel-de Siebenthal discrete series of $G_0$ is defined using $\\Delta^+$ in a manner analogous to the holomorphic discrete series.

Let $\\nu^*$ be the fundamental weight corresponding to $\\nu$ and $L_0$ be the centralizer in $K_0$ of the circle subgroup defined by $i\\nu^*$.

Note that $L_0 = K_0$, when $G_0/K_0$ is Hermitian symmetric. Otherwise, $L_0$ is a proper subgroup of $K_0$ and $K_0/L_0$ is an irreducible compact Hermitian symmetric space.

Let $G$ be the simply connected Lie group with Lie algebra $\\frak{g}$ and $K_0^* \\subset G$ be the dual of $K_0$ with respect to $L_0$ (or, the image of $L_0$ in $G$).

Then $K_0^*/L_0$ is an irreducible non-compact Hermitian symmetric space dual to $K_0/L_0$.

In this talk, to each Borel-de Siebenthal discrete series of $G_0$, a holomorphic discrete series of $K_0^*$ will be associated and occurrence of common $L_0$-types in both the series will be discussed.

Let $G_0$ be a simply connected non-compact real simple Lie group with maximal compact subgroup $K_0$.

Let $T_0\\subset K_0$ be a Cartan subgroup of $K_0$ as well as of $G_0$. So $G_0$ has discrete series representations.

Denote by $\\frak{g}, \\frak{k},$ and $\\frak{t}$, the

complexifications of the Lie algebras $\\frak{g}_0, \\frak{k}_0$ and $\\frak{t}_0$ of $G_0, K_0$ and $T_0$ respectively.

There exists a positive root system $\\Delta^+$ of $\\frak{g}$ with respect to $\\frak{t}$, known as the Borel-de Siebenthal positive system for which there is exactly one non-compact simple root, denoted $\\nu$. Let $\\mu$ denote the highest root.

If $G_0/K_0$ is Hermitian symmetric, then $\\nu$ has coefficient $1$ in $\\mu$ and one can define holomorphic discrete series representation of $G_0$ using $\\Delta^+$.

If $G_0/K_0$ is not Hermitian symmetric, the coefficient of $\\nu$ in the highest root $\\mu$ is $2$.

In this case, Borel-de Siebenthal discrete series of $G_0$ is defined using $\\Delta^+$ in a manner analogous to the holomorphic discrete series.

Let $\\nu^*$ be the fundamental weight corresponding to $\\nu$ and $L_0$ be the centralizer in $K_0$ of the circle subgroup defined by $i\\nu^*$.

Note that $L_0 = K_0$, when $G_0/K_0$ is Hermitian symmetric. Otherwise, $L_0$ is a proper subgroup of $K_0$ and $K_0/L_0$ is an irreducible compact Hermitian symmetric space.

Let $G$ be the simply connected Lie group with Lie algebra $\\frak{g}$ and $K_0^* \\subset G$ be the dual of $K_0$ with respect to $L_0$ (or, the image of $L_0$ in $G$).

Then $K_0^*/L_0$ is an irreducible non-compact Hermitian symmetric space dual to $K_0/L_0$.

In this talk, to each Borel-de Siebenthal discrete series of $G_0$, a holomorphic discrete series of $K_0^*$ will be associated and occurrence of common $L_0$-types in both the series will be discussed.

**Dipendra Prasad 氏**(Tata Institute of Fundamental Research) 16:50-17:40Branching laws and the local Langlands correspondence (ENGLISH)

[ 講演概要 ]

The decomposition of a representation of a group when restricted to a subgroup is an important problem well-studied for finite and compact Lie groups, and continues to be of much contemporary interest in the context of real and $p$-adic groups. We will survey some of the questions that have recently been considered drawing analogy with Compact Lie groups, and what it suggests in the context of real and $p$-adic groups via what is called the local Langlands correspondence.

The decomposition of a representation of a group when restricted to a subgroup is an important problem well-studied for finite and compact Lie groups, and continues to be of much contemporary interest in the context of real and $p$-adic groups. We will survey some of the questions that have recently been considered drawing analogy with Compact Lie groups, and what it suggests in the context of real and $p$-adic groups via what is called the local Langlands correspondence.

### 2013年10月29日(火)

16:30-18:00 数理科学研究科棟(駒場) 126号室

直交群の無重複表現の幾何と可視的な作用 (JAPANESE)

**田中雄一郎 氏**(東京大学大学院数理科学研究科)直交群の無重複表現の幾何と可視的な作用 (JAPANESE)

[ 講演概要 ]

For a connected compact simple Lie group of type B or D,

we find pairs $(V_{1},V_{2})$ of irreducible representations of G such that the tensor product representation $V_{1}¥otimes V_{2}$ is multiplicity-free by a geometric consideration based on

a notion of visible actions on complex manifolds,

introduced by T. Kobayashi. The pairs we find exhaust

all the multiplicity-free pairs by an earlier

combinatorial classification due to Stembridge.

For a connected compact simple Lie group of type B or D,

we find pairs $(V_{1},V_{2})$ of irreducible representations of G such that the tensor product representation $V_{1}¥otimes V_{2}$ is multiplicity-free by a geometric consideration based on

a notion of visible actions on complex manifolds,

introduced by T. Kobayashi. The pairs we find exhaust

all the multiplicity-free pairs by an earlier

combinatorial classification due to Stembridge.

### 2013年10月22日(火)

17:00-18:00 数理科学研究科棟(駒場) 126号室

Representation Theory and Microlocal Analysis (ENGLISH)

**Benjamin Harris 氏**(Louisiana State University (USA))Representation Theory and Microlocal Analysis (ENGLISH)

[ 講演概要 ]

Suppose $H\\subset K$ are compact, connected Lie groups, and suppose $\\tau$ is an irreducible, unitary representation of $H$. In 1979, Kashiwara and Vergne proved a simple asymptotic formula for the decomposition of $Ind_H^K\\tau$ by microlocally studying the regularity of vectors in this representation, thought of as vector valued functions on $K$. In 1998, Kobayashi proved a powerful criterion for the discrete decomposability of an irreducible, unitary representation $\\pi$ of a reductive Lie group $G$ when restricted to a reductive subgroup $H$. One of his key ideas was to restrict $\\pi$ to a representation of a maximal compact subgroup $K\\subset G$, view $\\pi$ as a subrepresentation of $L^2(K)$, and then use ideas similar to those developed by Kashiwara and Vergne.

In a recent preprint the speaker wrote with Hongyu He and Gestur Olafsson, the authors consider the possibility of studying induction and restriction to a reductive Lie group $G$ by microlocally studying the regularity of the matrix coefficients of (possibly reducible) unitary representations of $G$, viewed as continuous functions on the (possibly noncompact) Lie group $G$. In this talk, we will outline the main results of this paper and give additional conjectures.

Suppose $H\\subset K$ are compact, connected Lie groups, and suppose $\\tau$ is an irreducible, unitary representation of $H$. In 1979, Kashiwara and Vergne proved a simple asymptotic formula for the decomposition of $Ind_H^K\\tau$ by microlocally studying the regularity of vectors in this representation, thought of as vector valued functions on $K$. In 1998, Kobayashi proved a powerful criterion for the discrete decomposability of an irreducible, unitary representation $\\pi$ of a reductive Lie group $G$ when restricted to a reductive subgroup $H$. One of his key ideas was to restrict $\\pi$ to a representation of a maximal compact subgroup $K\\subset G$, view $\\pi$ as a subrepresentation of $L^2(K)$, and then use ideas similar to those developed by Kashiwara and Vergne.

In a recent preprint the speaker wrote with Hongyu He and Gestur Olafsson, the authors consider the possibility of studying induction and restriction to a reductive Lie group $G$ by microlocally studying the regularity of the matrix coefficients of (possibly reducible) unitary representations of $G$, viewed as continuous functions on the (possibly noncompact) Lie group $G$. In this talk, we will outline the main results of this paper and give additional conjectures.

### 2013年04月30日(火)

16:30-18:00 数理科学研究科棟(駒場) 126号室

The homomorphisms between scalar generalized Verma modules of gl(n,C) with regular infinitesimal characters (JAPANESE)

**松本久義 氏**(東京大学大学院数理科学研究科)The homomorphisms between scalar generalized Verma modules of gl(n,C) with regular infinitesimal characters (JAPANESE)

[ 講演概要 ]

We will explain the classification of the homomorphisms between scalar generalized Verma modules of gl(n,C) with regular infinitesimal characters. In fact, they are compositions of elementary homomorphisms. The main ingredient of our proof is the translation principle in the mediocre region.

We will explain the classification of the homomorphisms between scalar generalized Verma modules of gl(n,C) with regular infinitesimal characters. In fact, they are compositions of elementary homomorphisms. The main ingredient of our proof is the translation principle in the mediocre region.

### 2013年04月16日(火)

16:30-18:30 数理科学研究科棟(駒場) 126号室

Non-standard models for small representations of GL(n,R) (ENGLISH)

Degenerate principal series of symplectic groups (ENGLISH)

**Michael Pevzner 氏**(Reims University) 16:30-17:30Non-standard models for small representations of GL(n,R) (ENGLISH)

[ 講演概要 ]

We shall present new models for some parabolically induced

unitary representations of the real general linear group which involve Weyl symbolic calculus and furnish very efficient tools in studying branching laws for such representations.

We shall present new models for some parabolically induced

unitary representations of the real general linear group which involve Weyl symbolic calculus and furnish very efficient tools in studying branching laws for such representations.

**Pierre Clare 氏**(Penn. State University, USA) 17:30-18:30Degenerate principal series of symplectic groups (ENGLISH)

[ 講演概要 ]

We will discuss properties of representations of symplectic groups induced from maximal parabolic subgroups of Heisenberg type, including K-types formulas, expressions of intertwining operators and the study of their spectrum.

We will discuss properties of representations of symplectic groups induced from maximal parabolic subgroups of Heisenberg type, including K-types formulas, expressions of intertwining operators and the study of their spectrum.

### 2013年04月09日(火)

16:30-18:00 数理科学研究科棟(駒場) 126号室

A characterization of non-tube type Hermitian symmetric spaces by visible actions

(JAPANESE)

**笹木集夢 氏**(東海大学)A characterization of non-tube type Hermitian symmetric spaces by visible actions

(JAPANESE)

[ 講演概要 ]

We consider a non-symmetric complex Stein manifold D

which is realized as a line bundle over the complexification of a non-compact irreducible Hermitian symmetric space G/K.

In this talk, we will explain that the compact group action on D is strongly visible in the sense of Toshiyuki Kobayashi if and only if G/K is of non-tube type.

In particular, we focus on our construction of slice which meets every orbit in D from the viewpoint of group theory, namely,

we find an A-part of a generalized Cartan decomposition for homogeneous space D.

We note that our choice of A-part is an abelian.

We consider a non-symmetric complex Stein manifold D

which is realized as a line bundle over the complexification of a non-compact irreducible Hermitian symmetric space G/K.

In this talk, we will explain that the compact group action on D is strongly visible in the sense of Toshiyuki Kobayashi if and only if G/K is of non-tube type.

In particular, we focus on our construction of slice which meets every orbit in D from the viewpoint of group theory, namely,

we find an A-part of a generalized Cartan decomposition for homogeneous space D.

We note that our choice of A-part is an abelian.

### 2013年04月02日(火)

16:30-18:00 数理科学研究科棟(駒場) 126号室

Zuckerman導来関手加群の離散的分岐則 (JAPANESE)

**大島芳樹 氏**(Kavli IPMU, the University of Tokyo)Zuckerman導来関手加群の離散的分岐則 (JAPANESE)

[ 講演概要 ]

We consider the restriction of Zuckerman's derived functor modules with respect to symmetric pairs of real reductive groups. When they are discretely decomposable, explicit formulas for the branching laws are obtained by using a realization as D-module on the flag variety and the generalized BGG resolution. In this talk we would like to illustrate how to derive the formulas with a few examples.

We consider the restriction of Zuckerman's derived functor modules with respect to symmetric pairs of real reductive groups. When they are discretely decomposable, explicit formulas for the branching laws are obtained by using a realization as D-module on the flag variety and the generalized BGG resolution. In this talk we would like to illustrate how to derive the formulas with a few examples.

### 2013年02月05日(火)

17:30-19:00 数理科学研究科棟(駒場) 126号室

Dunkl processes assciated with dihedral systems, II (ENGLISH)

**Nizar Demni 氏**(Université de Rennes 1)Dunkl processes assciated with dihedral systems, II (ENGLISH)

[ 講演概要 ]

I'll focus on dihedral systems and its semi group density. I'll show how one can write down this density using probabilistic techniques and give some interpretation using spherical harmonics. I'll also present some results attempting to get a close formula for the density: the main difficulty comes then from the inversion (in composition sense) of Tchebycheff polynomials of the first kind in some neighborhood. Finally, I'll display expressions through known special functions for even dihedral groups, and the unexplained connection between the obtained formulas and those of Ben Said-Kobayashi-Orsted.

I'll focus on dihedral systems and its semi group density. I'll show how one can write down this density using probabilistic techniques and give some interpretation using spherical harmonics. I'll also present some results attempting to get a close formula for the density: the main difficulty comes then from the inversion (in composition sense) of Tchebycheff polynomials of the first kind in some neighborhood. Finally, I'll display expressions through known special functions for even dihedral groups, and the unexplained connection between the obtained formulas and those of Ben Said-Kobayashi-Orsted.

### 2013年02月04日(月)

17:30-19:00 数理科学研究科棟(駒場) 126号室

いつもと曜日・時刻が違います

Dunkl processes assciated with dihedral systems, I (ENGLISH)

いつもと曜日・時刻が違います

**Nizar Demni 氏**(Université de Rennes 1)Dunkl processes assciated with dihedral systems, I (ENGLISH)

[ 講演概要 ]

I'll first give a brief and needed account on root systems and finite reflection groups. Then, I'll introduce Dunkl operators and give some properties. Once I'll do, I'll introduce Dunkl processes and their continuous components, so-called radial Dunkl processes. The latter generalize eigenvalues processes of some matrix-valued processes and reduces to reflected Brownian motion in Weyl chambers. Besides, Brownian motion in Weyl chambers corresponds to all multiplicity values equal one are constructed from a Brownian motion killed when it first hits the boundary of the Weyl chamber using the unique positive harmonic function (up to a constant) on the Weyl chamber. In the analytic side, determinantal formulas appear and are related to harmonic analysis on the Gelfand pair (Gl(n,C), U(n)). This is in agreement on the one side with the so-called reflection principle in stochastic processes theory and matches on the other side the so-called shift principle introduced by E. Opdam. Finally, I'll discuss the spectacular result of Biane-Bougerol-O'connell yielding to a Duistermaat-Heckman distribution for non crystallographic systems.

I'll first give a brief and needed account on root systems and finite reflection groups. Then, I'll introduce Dunkl operators and give some properties. Once I'll do, I'll introduce Dunkl processes and their continuous components, so-called radial Dunkl processes. The latter generalize eigenvalues processes of some matrix-valued processes and reduces to reflected Brownian motion in Weyl chambers. Besides, Brownian motion in Weyl chambers corresponds to all multiplicity values equal one are constructed from a Brownian motion killed when it first hits the boundary of the Weyl chamber using the unique positive harmonic function (up to a constant) on the Weyl chamber. In the analytic side, determinantal formulas appear and are related to harmonic analysis on the Gelfand pair (Gl(n,C), U(n)). This is in agreement on the one side with the so-called reflection principle in stochastic processes theory and matches on the other side the so-called shift principle introduced by E. Opdam. Finally, I'll discuss the spectacular result of Biane-Bougerol-O'connell yielding to a Duistermaat-Heckman distribution for non crystallographic systems.

### 2013年01月22日(火)

16:30-18:00 数理科学研究科棟(駒場) 126号室

Representation theory of finite W-algebras (ENGLISH)

**Simon Goodwin 氏**(Birmingham University)Representation theory of finite W-algebras (ENGLISH)

[ 講演概要 ]

There has been a great deal of recent research interest in finite W-algebras motivated by important connection with primitive ideals of universal enveloping algebras and applications in mathematical physics.

There have been significant breakthroughs in the rerpesentation theory of finite W-algebras due to the research of a variety of mathematicians.

In this talk, we will give an overview of the representation theory of finite W-algebras focussing on W-algebras associated to classical Lie algebras (joint with J. Brown) and W-algebras associated to general linear Lie superalgebras (joint with J. Brown and J. Brundan).

There has been a great deal of recent research interest in finite W-algebras motivated by important connection with primitive ideals of universal enveloping algebras and applications in mathematical physics.

There have been significant breakthroughs in the rerpesentation theory of finite W-algebras due to the research of a variety of mathematicians.

In this talk, we will give an overview of the representation theory of finite W-algebras focussing on W-algebras associated to classical Lie algebras (joint with J. Brown) and W-algebras associated to general linear Lie superalgebras (joint with J. Brown and J. Brundan).

### 2013年01月08日(火)

16:30-18:00 数理科学研究科棟(駒場) 126号室

On the group of holomorphic and anti-holomorphic transformations

of a compact Hermitian symmetric space and the $G$-structure (JAPANESE)

**金行壮二 氏**(上智大学(名誉教授))On the group of holomorphic and anti-holomorphic transformations

of a compact Hermitian symmetric space and the $G$-structure (JAPANESE)

[ 講演概要 ]

Let $M$ be a compact irreducible Hermitian symmetric space. We determine the full group of holomorphic and anti-holomorphic transformations of $M$. Also we characterize that full group as the automorphism group of the $G$-structure on $M$, called a generalized conformal structure.

Let $M$ be a compact irreducible Hermitian symmetric space. We determine the full group of holomorphic and anti-holomorphic transformations of $M$. Also we characterize that full group as the automorphism group of the $G$-structure on $M$, called a generalized conformal structure.

### 2012年12月11日(火)

16:30-18:00 数理科学研究科棟(駒場) 126号室

Affine Springer fibers of type A and combinatorics of diagonal

coinvariants

Affine Springer fibers of type A and combinatorics of diagonal

coinvariants (JAPANESE)

**疋田辰之 氏**(京都大学大学院理学研究科)Affine Springer fibers of type A and combinatorics of diagonal

coinvariants

Affine Springer fibers of type A and combinatorics of diagonal

coinvariants (JAPANESE)

[ 講演概要 ]

We introduce certain filtrations on the homology of

affine Springer fibers of type A and give combinatorial formulas for the bigraded Frobenius series of the associated graded modules.

The results are essentially given by generalizations of the symmetric function introduced by Haglund, Haiman, Loehr, Remmel, and Ulyanov which is conjectured to coincide with the bigraded Frobenius series of the ring of diagonal coinvariants.

We introduce certain filtrations on the homology of

affine Springer fibers of type A and give combinatorial formulas for the bigraded Frobenius series of the associated graded modules.

The results are essentially given by generalizations of the symmetric function introduced by Haglund, Haiman, Loehr, Remmel, and Ulyanov which is conjectured to coincide with the bigraded Frobenius series of the ring of diagonal coinvariants.

### 2012年11月29日(木)

16:30-17:30 数理科学研究科棟(駒場) 122号室

いつもと部屋が違います

On a relation between certain character values of symmetric groups (JAPANESE)

いつもと部屋が違います

**渡部正樹 氏**(東京大学大学院数理科学研究科)On a relation between certain character values of symmetric groups (JAPANESE)

[ 講演概要 ]

We present a relation of new kind between character values of

symmetric groups which explains a curious phenomenon in character

tables of symmetric groups. Similar relations for characters of

Brauer and walled Brauer algebras and projective characters of

symmetric groups are also presented.

We present a relation of new kind between character values of

symmetric groups which explains a curious phenomenon in character

tables of symmetric groups. Similar relations for characters of

Brauer and walled Brauer algebras and projective characters of

symmetric groups are also presented.

### 2012年11月27日(火)

16:30-18:00 数理科学研究科棟(駒場) 126号室

旗多様体のケーラー偏極の実偏極への収束 (JAPANESE)

**今野宏 氏**(東京大学大学院数理科学研究科)旗多様体のケーラー偏極の実偏極への収束 (JAPANESE)

[ 講演概要 ]

In this talk we will discuss geometric quantization of a flag manifold. In particular, we construct a family of complex structures on a flag manifold that converge 'at the quantum level' to the real polarization coming from the Gelfand-Cetlin integrable system.

Our construction is based on a toric degeneration of flag varieties and a deformation of K¥"ahler structure on toric varieties by symplectic potentials.

This is a joint work with Mark Hamilton.

In this talk we will discuss geometric quantization of a flag manifold. In particular, we construct a family of complex structures on a flag manifold that converge 'at the quantum level' to the real polarization coming from the Gelfand-Cetlin integrable system.

Our construction is based on a toric degeneration of flag varieties and a deformation of K¥"ahler structure on toric varieties by symplectic potentials.

This is a joint work with Mark Hamilton.

### 2012年11月20日(火)

16:30-17:30 数理科学研究科棟(駒場) 126号室

On the geometry of discontinuous subgroups acting on some homogeneous spaces (ENGLISH)

**Ali Baklouti 氏**(Sfax University)On the geometry of discontinuous subgroups acting on some homogeneous spaces (ENGLISH)

[ 講演概要 ]

Let $G$ be a Lie group, $H$ a closed subgroup of $G$ and \\Gamma$ a discontinuous subgroup for the homogeneous space $G/H$. I first introduce the deformation space ${\\mathcal{T}}^{K_o}(\\Gamma, G, H)$ of the action of $\\Gamma$ on $G/H$ in the sense of Kobayashi and some of its refined versions, namely the Clifford--Klein space of deformations of the form ${\\mathcal{X}}=\\Gamma \\backslash G/H$. The deformation space ${\\mathcal{T}}^{G_o}(\\Gamma, G,H)$ of marked $(G,H)$-structures on ${\\mathcal{X}}$ in the sense of Goldman is also introduced. As an important motivation, I will explain the connection between the spaces ${\\mathcal{T}}^{K_o}(\\Gamma, G, H)$ and ${\\mathcal{T}}^{G_o}(\\Gamma, G, H)$ and study some of their topological features, namely the rigidity in the sense of Selberg--Weil--Kobayashi and the stability in the sense of Kobayashi--Nasrin. The latter appears to be of major interest to write down the connection explicitly.

Let $G$ be a Lie group, $H$ a closed subgroup of $G$ and \\Gamma$ a discontinuous subgroup for the homogeneous space $G/H$. I first introduce the deformation space ${\\mathcal{T}}^{K_o}(\\Gamma, G, H)$ of the action of $\\Gamma$ on $G/H$ in the sense of Kobayashi and some of its refined versions, namely the Clifford--Klein space of deformations of the form ${\\mathcal{X}}=\\Gamma \\backslash G/H$. The deformation space ${\\mathcal{T}}^{G_o}(\\Gamma, G,H)$ of marked $(G,H)$-structures on ${\\mathcal{X}}$ in the sense of Goldman is also introduced. As an important motivation, I will explain the connection between the spaces ${\\mathcal{T}}^{K_o}(\\Gamma, G, H)$ and ${\\mathcal{T}}^{G_o}(\\Gamma, G, H)$ and study some of their topological features, namely the rigidity in the sense of Selberg--Weil--Kobayashi and the stability in the sense of Kobayashi--Nasrin. The latter appears to be of major interest to write down the connection explicitly.

### 2012年11月13日(火)

16:30-18:00 数理科学研究科棟(駒場) 126号室

Tight maps, a classification (ENGLISH)

**Oskar Hamlet 氏**(Chalmers University)Tight maps, a classification (ENGLISH)

[ 講演概要 ]

Tight maps/homomorphisms were introduced during the study of rigidity properties of surface groups in Hermitian Lie groups. In this talk I'll discuss the properties of tight maps, their connection to rigidity theory and my work classifying them.

Tight maps/homomorphisms were introduced during the study of rigidity properties of surface groups in Hermitian Lie groups. In this talk I'll discuss the properties of tight maps, their connection to rigidity theory and my work classifying them.

### 2012年11月06日(火)

16:30-17:30 数理科学研究科棟(駒場) 126号室

An explicit construction of spherical designs on S^3 (JAPANESE)

**奥田隆幸 氏**(東京大学)An explicit construction of spherical designs on S^3 (JAPANESE)

[ 講演概要 ]

The existence of spherical t-designs on S^d for any t and d are proved by Seymour--Zaslavsky in 1984.

However, explicit constructions of spherical designs were not known for d > 2 and large t.

In this talk, for a given spherical t-design Y on S^2, we give an

algorithm to make a spherical 2t-design X on S^3 which maps Y by a Hopf map. In particular, by combining with the results of Kuperberg in 2005, we have an explicit construction of spherical t-designs on S^3 for any t.

The existence of spherical t-designs on S^d for any t and d are proved by Seymour--Zaslavsky in 1984.

However, explicit constructions of spherical designs were not known for d > 2 and large t.

In this talk, for a given spherical t-design Y on S^2, we give an

algorithm to make a spherical 2t-design X on S^3 which maps Y by a Hopf map. In particular, by combining with the results of Kuperberg in 2005, we have an explicit construction of spherical t-designs on S^3 for any t.

### 2012年07月24日(火)

16:30-18:00 数理科学研究科棟(駒場) 126号室

The Dynkin index and conformally invariant systems of Heisenberg parabolic type (ENGLISH)

**久保利久 氏**(東京大学 大学院 数理科学研究科)The Dynkin index and conformally invariant systems of Heisenberg parabolic type (ENGLISH)

[ 講演概要 ]

Heisenberg parabolic subalgebraから構築されたconformally invariant

systemに関する二つの定数について考察する。それらの定数はそのconformally

invariant systemを構築する際に重要な役割を果たしたが、ある二つの式の比率

として定義されただけで具体的な式などは表されなかった。本講演ではそれらの

具体的かつ一様な式をDynkin indexを交えて紹介する。

Heisenberg parabolic subalgebraから構築されたconformally invariant

systemに関する二つの定数について考察する。それらの定数はそのconformally

invariant systemを構築する際に重要な役割を果たしたが、ある二つの式の比率

として定義されただけで具体的な式などは表されなかった。本講演ではそれらの

具体的かつ一様な式をDynkin indexを交えて紹介する。

### 2012年07月17日(火)

17:00-18:30 数理科学研究科棟(駒場) 126号室

Dirac induction for graded affine Hecke algebras (ENGLISH)

**Eric Opdam 氏**(Universiteit van Amsterdam)Dirac induction for graded affine Hecke algebras (ENGLISH)

[ 講演概要 ]

In recent joint work with Dan Ciubotaru and Peter Trapa we

constructed a model for the discrete series representations of graded affine Hecke algebras as the index of a Dirac operator.

We discuss the K-theoretic meaning of this result, and the remarkable relation between elliptic character theory of a Weyl group and the ordinary character theory of its Pin cover.

In recent joint work with Dan Ciubotaru and Peter Trapa we

constructed a model for the discrete series representations of graded affine Hecke algebras as the index of a Dirac operator.

We discuss the K-theoretic meaning of this result, and the remarkable relation between elliptic character theory of a Weyl group and the ordinary character theory of its Pin cover.

### 2012年06月12日(火)

16:30-18:00 数理科学研究科棟(駒場) 126号室

Conformally invariant systems of differential operators of non-Heisenberg parabolic type (ENGLISH)

**久保利久 氏**(東京大学大学院数理科学研究科)Conformally invariant systems of differential operators of non-Heisenberg parabolic type (ENGLISH)

[ 講演概要 ]

Minkowski space上のwave operatorはconformally invariant operatorの典型的な例である。

近年、Barchini-Kable-Zierauによって1つのdifferential operatorの

conformal invarianceがそのsystemに一般化された (conformally invariant systems)。

このセミナーではmaximal non-Heisenberg parabolicを使って、

その様なsecond order differential operatorのsystemを作りたい。

またconformally invariant systemは、あるgeneralized Verma module間のhomomorphismを誘導するが、もし時間が許せばそれについても述べたい。

Minkowski space上のwave operatorはconformally invariant operatorの典型的な例である。

近年、Barchini-Kable-Zierauによって1つのdifferential operatorの

conformal invarianceがそのsystemに一般化された (conformally invariant systems)。

このセミナーではmaximal non-Heisenberg parabolicを使って、

その様なsecond order differential operatorのsystemを作りたい。

またconformally invariant systemは、あるgeneralized Verma module間のhomomorphismを誘導するが、もし時間が許せばそれについても述べたい。

### 2012年06月05日(火)

16:30-18:00 数理科学研究科棟(駒場) 126号室

GCOE lectures

Random walk on reductive groups (ENGLISH)

GCOE lectures

**Yves Benoist 氏**(CNRS and Orsay)Random walk on reductive groups (ENGLISH)

[ 講演概要 ]

The asymptotic behavior of the sum of real numbers chosen independantly with same probability law is controled by many classical theorems: Law of Large Numbers, Central Limit Theorem, Law of Iterated Logarithm, Local Limit Theorem, Large deviation Principle, 0-1 Law,... In these introductory talks I will recall these classical results and explain their analogs for products of matrices chosen independantly with same probability law, when the action of the support of the law is semisimple. We will see that the dynamics of the corresponding action on the flag variety is a crucial tool for studying these non-commutative random walks.

The asymptotic behavior of the sum of real numbers chosen independantly with same probability law is controled by many classical theorems: Law of Large Numbers, Central Limit Theorem, Law of Iterated Logarithm, Local Limit Theorem, Large deviation Principle, 0-1 Law,... In these introductory talks I will recall these classical results and explain their analogs for products of matrices chosen independantly with same probability law, when the action of the support of the law is semisimple. We will see that the dynamics of the corresponding action on the flag variety is a crucial tool for studying these non-commutative random walks.