## 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**

### 2011/11/29

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Symmetries, (their) deformations, and physics: some perspectives and open problems from half a century of personal experience (ENGLISH)

**Daniel Sternheimer**(Rikkyo Univertiry and Université de Bourgogne)Symmetries, (their) deformations, and physics: some perspectives and open problems from half a century of personal experience (ENGLISH)

[ Abstract ]

This is a flexible general framework, based on quite a number of papers, some of which are reviewed in:

MR2285047 (2008c:53079) Sternheimer, Daniel. The geometry of space-time and its deformations from a physical perspective. From geometry to quantum mechanics, 287–301, Progr. Math., 252, Birkhäuser Boston, Boston, MA, 2007

http://monge.u-bourgogne.fr/d.sternh/papers/PiMOmori-DS.pdf

This is a flexible general framework, based on quite a number of papers, some of which are reviewed in:

MR2285047 (2008c:53079) Sternheimer, Daniel. The geometry of space-time and its deformations from a physical perspective. From geometry to quantum mechanics, 287–301, Progr. Math., 252, Birkhäuser Boston, Boston, MA, 2007

http://monge.u-bourgogne.fr/d.sternh/papers/PiMOmori-DS.pdf

### 2011/11/22

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Smallest complex nilpotent orbit with real points (JAPANESE)

**Takayuki Okuda**(東京大学大学院 数理科学研究科)Smallest complex nilpotent orbit with real points (JAPANESE)

[ Abstract ]

Let $\\mathfrak{g}$ be a non-compact simple Lie algebra with no complex

structures.

In this talk, we show that there exists a complex nilpotent orbit

$\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$ in

$\\mathfrak{g}_\\mathbb{C}$ ($:=\\mathfrak{g} \\otimes \\mathbb{C}$)

containing all of real nilpotent orbits in $\\mathfrak{g}$ of minimal

positive dimension.

For many $\\mathfrak{g}$, the orbit

$\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$ is just the

complex minimal nilpotent orbit in $\\mathfrak{g}_\\mathbb{C}$.

However, for the cases where $\\mathfrak{g}$ is isomorphic to

$\\mathfrak{su}^*(2k)$, $\\mathfrak{so}(n-1,1)$, $\\mathfrak{sp}(p,q)$,

$\\mathfrak{e}_{6(-26)}$ or $\\mathfrak{f}_{4(-20)}$,

the orbit $\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$ is not

the complex minimal nilpotent orbit in $\\mathfrak{g}_\\mathbb{C}$.

We also determine $\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$

by describing the weighted Dynkin diagrams of these for such cases.

Let $\\mathfrak{g}$ be a non-compact simple Lie algebra with no complex

structures.

In this talk, we show that there exists a complex nilpotent orbit

$\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$ in

$\\mathfrak{g}_\\mathbb{C}$ ($:=\\mathfrak{g} \\otimes \\mathbb{C}$)

containing all of real nilpotent orbits in $\\mathfrak{g}$ of minimal

positive dimension.

For many $\\mathfrak{g}$, the orbit

$\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$ is just the

complex minimal nilpotent orbit in $\\mathfrak{g}_\\mathbb{C}$.

However, for the cases where $\\mathfrak{g}$ is isomorphic to

$\\mathfrak{su}^*(2k)$, $\\mathfrak{so}(n-1,1)$, $\\mathfrak{sp}(p,q)$,

$\\mathfrak{e}_{6(-26)}$ or $\\mathfrak{f}_{4(-20)}$,

the orbit $\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$ is not

the complex minimal nilpotent orbit in $\\mathfrak{g}_\\mathbb{C}$.

We also determine $\\mathcal{O}^{G_\\mathbb{C}}_{\\text{min},\\mathfrak{g}}$

by describing the weighted Dynkin diagrams of these for such cases.

### 2011/11/15

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

Categorification of cluster algebras arising from unipotent subgroups of non-simply laced Lie groups (ENGLISH)

**Laurant Demonet**(Nagoya University)Categorification of cluster algebras arising from unipotent subgroups of non-simply laced Lie groups (ENGLISH)

[ Abstract ]

We introduce an abstract framework to categorify some antisymetrizable cluster algebras by using actions of finite groups on stably 2-Calabi-Yau exact categories. We introduce the notion of the equivariant category and, with similar technics as in [K], [CK], [GLS1], [GLS2], [DK], [FK], [P], we construct some examples of such categorifications. For example, if we let Z/2Z act on the category of representations of the preprojective algebra of type A2n-1 via the only non trivial action on the diagram, we obtain the cluster structure on the coordinate ring of the maximal unipotent subgroup of the semi-simple Lie group of type Bn [D]. Hence, we can get relations between the cluster algebras categorified by some exact subcategories of these two categories. We also prove by the same methods as in [FK] a conjecture of Fomin and Zelevinsky stating that the cluster monomials are linearly independent.

References

[CK] P. Caldero, B. Keller, From triangulated categories to cluster algebras, Invent. Math. 172 (2008), no. 1, 169--211.

[DK] R. Dehy, B. Keller, On the combinatorics of rigid objects in 2-Calabi-Yau categories, arXiv: 0709.0882.

[D] L. Demonet, Cluster algebras and preprojective algebras: the non simply-laced case, C. R. Acad. Sci. Paris, Ser. I 346 (2008), 379--384.

[FK] C. Fu, B. Keller, On cluster algebras with coefficients and 2-Calabi-Yau categories, arXiv: 0710.3152.

[GLS1] C. Geiss, B. Leclerc, J. Schröer, Rigid modules over preprojective algebras, Invent. Math. 165 (2006), no. 3, 589--632.

[GLS2] C. Geiss, B. Leclerc, J. Schröer, Cluster algebra structures and semicanoncial bases for unipotent groups, arXiv: math/0703039.

[K] B. Keller, Categorification of acyclic cluster algebras: an introduction, arXiv: 0801.3103.

[P] Y. Palu, Cluster characters for triangulated 2-Calabi--Yau categories, arXiv: math/0703540.

We introduce an abstract framework to categorify some antisymetrizable cluster algebras by using actions of finite groups on stably 2-Calabi-Yau exact categories. We introduce the notion of the equivariant category and, with similar technics as in [K], [CK], [GLS1], [GLS2], [DK], [FK], [P], we construct some examples of such categorifications. For example, if we let Z/2Z act on the category of representations of the preprojective algebra of type A2n-1 via the only non trivial action on the diagram, we obtain the cluster structure on the coordinate ring of the maximal unipotent subgroup of the semi-simple Lie group of type Bn [D]. Hence, we can get relations between the cluster algebras categorified by some exact subcategories of these two categories. We also prove by the same methods as in [FK] a conjecture of Fomin and Zelevinsky stating that the cluster monomials are linearly independent.

References

[CK] P. Caldero, B. Keller, From triangulated categories to cluster algebras, Invent. Math. 172 (2008), no. 1, 169--211.

[DK] R. Dehy, B. Keller, On the combinatorics of rigid objects in 2-Calabi-Yau categories, arXiv: 0709.0882.

[D] L. Demonet, Cluster algebras and preprojective algebras: the non simply-laced case, C. R. Acad. Sci. Paris, Ser. I 346 (2008), 379--384.

[FK] C. Fu, B. Keller, On cluster algebras with coefficients and 2-Calabi-Yau categories, arXiv: 0710.3152.

[GLS1] C. Geiss, B. Leclerc, J. Schröer, Rigid modules over preprojective algebras, Invent. Math. 165 (2006), no. 3, 589--632.

[GLS2] C. Geiss, B. Leclerc, J. Schröer, Cluster algebra structures and semicanoncial bases for unipotent groups, arXiv: math/0703039.

[K] B. Keller, Categorification of acyclic cluster algebras: an introduction, arXiv: 0801.3103.

[P] Y. Palu, Cluster characters for triangulated 2-Calabi--Yau categories, arXiv: math/0703540.

### 2011/10/25

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

Localization of Cohomological Induction (ENGLISH)

**Yoshiki Oshima**(Graduate School of Mathematical Sciences, the University of Tokyo)Localization of Cohomological Induction (ENGLISH)

[ Abstract ]

Cohomological induction is defined for (g,K)-modules in an algebraic way and construct important representations such as (Harish-Chandra modules of) discrete series representations,

principal series representations and Zuckerman's modules of

semisimple Lie groups.

Hecht, Milicic, Schmid, and Wolf proved that modules induced from

one-dimensional representations of Borel subalgebra can be realized as D-modules on the flag variety.

In this talk, we show a similar result for modules induced from

more general representations.

Cohomological induction is defined for (g,K)-modules in an algebraic way and construct important representations such as (Harish-Chandra modules of) discrete series representations,

principal series representations and Zuckerman's modules of

semisimple Lie groups.

Hecht, Milicic, Schmid, and Wolf proved that modules induced from

one-dimensional representations of Borel subalgebra can be realized as D-modules on the flag variety.

In this talk, we show a similar result for modules induced from

more general representations.

### 2011/06/07

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Rigidity of group actions via invariant geometric structures

(JAPANESE)

**Masahiko Kanai**(the University of Tokyo)Rigidity of group actions via invariant geometric structures

(JAPANESE)

[ Abstract ]

It is a homomorphism into a FINITE dimensional Lie group that is concerned with in the classical RIGIDITY theorems such as those of Mostow and Margulis. In the meantime, differentiable GROUP ACTIONS for which we ask rigidity problems is a homomorphism into a diffeomorphism group, which is a typical example of INFINITE dimensional Lie groups. The purpose of the present talk is exhibiting several rigidity theorems for group actions in which I have been involved for years. Although quite a few fields of mathematics, such as ergodic theory, the theory of smooth dynamical systems, representation theory and so on, have made remarkable contributions to rigidity problems, I would rather emphasis geometric aspects: I would focus on those rigidity phenomenon for group actions that are observed by showing that the actions have INVARIANT GEOMETRIC STRUCTURES.

It is a homomorphism into a FINITE dimensional Lie group that is concerned with in the classical RIGIDITY theorems such as those of Mostow and Margulis. In the meantime, differentiable GROUP ACTIONS for which we ask rigidity problems is a homomorphism into a diffeomorphism group, which is a typical example of INFINITE dimensional Lie groups. The purpose of the present talk is exhibiting several rigidity theorems for group actions in which I have been involved for years. Although quite a few fields of mathematics, such as ergodic theory, the theory of smooth dynamical systems, representation theory and so on, have made remarkable contributions to rigidity problems, I would rather emphasis geometric aspects: I would focus on those rigidity phenomenon for group actions that are observed by showing that the actions have INVARIANT GEOMETRIC STRUCTURES.

### 2011/05/31

16:30-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

On character tables of association schemes based on attenuated

spaces (JAPANESE)

**Hirotake Kurihara**(Tohoku University)On character tables of association schemes based on attenuated

spaces (JAPANESE)

[ Abstract ]

An association scheme is a pair of a finite set $X$

and a set of relations $\\{R_i\\}_{0\\le i\\le d}$

on $X$ which satisfies several axioms of regularity.

The notion of association schemes is viewed as some axiomatized

properties of transitive permutation groups in terms of combinatorics, and also the notion of association schemes is regarded as a generalization of the subring of the group ring spanned by the conjugacy classes of finite groups.

Thus, the theory of association schemes had been developed in the

study of finite permutation groups and representation theory.

To determine the character tables of association schemes is an

important first step to a systematic study of association schemes, and is helpful toward the classification of those schemes.

In this talk, we determine the character tables of association schemes based on attenuated spaces.

These association schemes are obtained from subspaces of a given

dimension in attenuated spaces.

An association scheme is a pair of a finite set $X$

and a set of relations $\\{R_i\\}_{0\\le i\\le d}$

on $X$ which satisfies several axioms of regularity.

The notion of association schemes is viewed as some axiomatized

properties of transitive permutation groups in terms of combinatorics, and also the notion of association schemes is regarded as a generalization of the subring of the group ring spanned by the conjugacy classes of finite groups.

Thus, the theory of association schemes had been developed in the

study of finite permutation groups and representation theory.

To determine the character tables of association schemes is an

important first step to a systematic study of association schemes, and is helpful toward the classification of those schemes.

In this talk, we determine the character tables of association schemes based on attenuated spaces.

These association schemes are obtained from subspaces of a given

dimension in attenuated spaces.

### 2011/05/24

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

Properly discontinuous isometric group actions on inhomogeneous Lorentzian manifolds (JAPANESE)

**Jun-ichi Mukuno**(Nagoya University)Properly discontinuous isometric group actions on inhomogeneous Lorentzian manifolds (JAPANESE)

[ Abstract ]

If a homogeneous space $G/H$ is acted properly discontinuously

upon by a subgroup $\\Gamma$ of $G$ via the left action, the quotient space $\\Gamma \\backslash G/H$ is called a

Clifford--Klein form. In 1962, E. Calabi and L. Markus proved that there is no infinite subgroup of the Lorentz group $O(n+1, 1)$ whose left action on the de Sitter space $O(n+1, 1)/O(n, 1)$ is properly discontinuous.

It follows that a compact Clifford--Klein form of the de Sitter space never exists.

In this talk, we present a new extension of the theorem of E. Calabi and L. Markus to a certain class of Lorentzian manifolds that are not necessarily homogeneous.

If a homogeneous space $G/H$ is acted properly discontinuously

upon by a subgroup $\\Gamma$ of $G$ via the left action, the quotient space $\\Gamma \\backslash G/H$ is called a

Clifford--Klein form. In 1962, E. Calabi and L. Markus proved that there is no infinite subgroup of the Lorentz group $O(n+1, 1)$ whose left action on the de Sitter space $O(n+1, 1)/O(n, 1)$ is properly discontinuous.

It follows that a compact Clifford--Klein form of the de Sitter space never exists.

In this talk, we present a new extension of the theorem of E. Calabi and L. Markus to a certain class of Lorentzian manifolds that are not necessarily homogeneous.

### 2011/04/26

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Topological Blow-up (JAPANESE)

**Taro YOSHINO**(the University of Tokyo)Topological Blow-up (JAPANESE)

[ Abstract ]

Suppose that a Lie group $G$ acts on a manifold

$M$. The quotient space $X:=G\\backslash M$ is locally compact,

but not Hausdorff in general. Our aim is to understand

such a non-Hausdorff space $X$.

The space $X$ has the crack $S$. Rougly speaking, $S$ is

the causal subset of non-Hausdorffness of $X$, and especially

$X\\setminus S$ is Hausdorff.

We introduce the concept of `topological blow-up' as a `repair'

of the crack. The `repaired' space $\\tilde{X}$ is

locally compact and Hausdorff space containing $X\\setminus S$

as its open subset. Moreover, the original space $X$ can be

recovered from the pair of $(\\tilde{X}, S)$.

Suppose that a Lie group $G$ acts on a manifold

$M$. The quotient space $X:=G\\backslash M$ is locally compact,

but not Hausdorff in general. Our aim is to understand

such a non-Hausdorff space $X$.

The space $X$ has the crack $S$. Rougly speaking, $S$ is

the causal subset of non-Hausdorffness of $X$, and especially

$X\\setminus S$ is Hausdorff.

We introduce the concept of `topological blow-up' as a `repair'

of the crack. The `repaired' space $\\tilde{X}$ is

locally compact and Hausdorff space containing $X\\setminus S$

as its open subset. Moreover, the original space $X$ can be

recovered from the pair of $(\\tilde{X}, S)$.

### 2011/01/18

17:00-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Connections between Noncommutative Geometry and Lie groups

representations (ENGLISH)

**Pierre Clare**(Universite Orleans and the University of Tokyo)Connections between Noncommutative Geometry and Lie groups

representations (ENGLISH)

[ Abstract ]

One of the principles of Noncommutative Geometry is to study singular spaces that the tools of classical analysis like algebras of continuous functions fail to describe, replacing them by more general C*-algebras. After recalling fundamental facts about C*-algebras, Hilbert modules and group C*-algebras, we will present constructions and results aiming to understand principal series representations and Knapp-Stein theory in the noncommutative geometrical framework. Eventually we will explain the relationship between the analysis of reduced group C*-algebras and the computation of the Connes-Kasparov isomorphisms.

One of the principles of Noncommutative Geometry is to study singular spaces that the tools of classical analysis like algebras of continuous functions fail to describe, replacing them by more general C*-algebras. After recalling fundamental facts about C*-algebras, Hilbert modules and group C*-algebras, we will present constructions and results aiming to understand principal series representations and Knapp-Stein theory in the noncommutative geometrical framework. Eventually we will explain the relationship between the analysis of reduced group C*-algebras and the computation of the Connes-Kasparov isomorphisms.

### 2010/12/21

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

Some relation between the Weyl module and the crystal basis of the tensor product of fudamental representations (ENGLISH)

**Katsuyuki NAOI**(Graduate School of Mathematical Sciences, the University of Tokyo)Some relation between the Weyl module and the crystal basis of the tensor product of fudamental representations (ENGLISH)

### 2010/11/02

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

Twistor theory and the harmonic hull (ENGLISH)

**Michael Eastwood**(University of Adelaide)Twistor theory and the harmonic hull (ENGLISH)

[ Abstract ]

Harmonic functions are real-analytic and so automatically extend from being functions of real variables to being functions of complex variables. But how far do they extend? This question may be answered by twistor theory, the Penrose transform, and associated geometry. I shall base the constructions on a formula of Bateman from 1904. This is joint work with Feng Xu.

Harmonic functions are real-analytic and so automatically extend from being functions of real variables to being functions of complex variables. But how far do they extend? This question may be answered by twistor theory, the Penrose transform, and associated geometry. I shall base the constructions on a formula of Bateman from 1904. This is joint work with Feng Xu.

### 2010/10/26

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

Some instances of the reasonable effectiveness (and limitations) of symmetries and deformations in fundamental physics (ENGLISH)

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html

**Daniel Sternheimer**(Keio University and Institut de Mathematiques de Bourgogne)Some instances of the reasonable effectiveness (and limitations) of symmetries and deformations in fundamental physics (ENGLISH)

[ Abstract ]

In this talk we survey some applications of group theory and deformation theory (including quantization) in mathematical physics. We start with sketching applications of rotation and discrete groups representations in molecular physics (``dynamical" symmetry breaking in crystals, Racah-Flato-Kibler; chains of groups and symmetry breaking). These methods led to the use of ``classification Lie groups" (``internal symmetries") in particle physics. Their relation with space-time symmetries will be discussed. Symmetries are naturally deformed, which eventually brought to Flato's deformation philosophy and the realization that quantization can be viewed as a deformation, including the many avatars of deformation quantization (such as quantum groups and quantized spaces). Nonlinear representations of Lie groups can be viewed as deformations (of their linear part), with applications to covariant nonlinear evolution equations. Combining all these suggests an Ansatz based on Anti de Sitter space-time and group, a deformation of the Poincare group of Minkowski space-time, which could eventually be quantized, with possible implications in particle physics and cosmology. Prospects for future developments between mathematics and physics will be indicated.

[ Reference URL ]In this talk we survey some applications of group theory and deformation theory (including quantization) in mathematical physics. We start with sketching applications of rotation and discrete groups representations in molecular physics (``dynamical" symmetry breaking in crystals, Racah-Flato-Kibler; chains of groups and symmetry breaking). These methods led to the use of ``classification Lie groups" (``internal symmetries") in particle physics. Their relation with space-time symmetries will be discussed. Symmetries are naturally deformed, which eventually brought to Flato's deformation philosophy and the realization that quantization can be viewed as a deformation, including the many avatars of deformation quantization (such as quantum groups and quantized spaces). Nonlinear representations of Lie groups can be viewed as deformations (of their linear part), with applications to covariant nonlinear evolution equations. Combining all these suggests an Ansatz based on Anti de Sitter space-time and group, a deformation of the Poincare group of Minkowski space-time, which could eventually be quantized, with possible implications in particle physics and cosmology. Prospects for future developments between mathematics and physics will be indicated.

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html

### 2010/09/01

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Groups of Kac-Moody type (ENGLISH)

**Bernhard M\"uhlherr**(Justus-Liebig-Universit\"at Giessen)Groups of Kac-Moody type (ENGLISH)

[ Abstract ]

Groups of Kac-Moody type are natural generalizations of Kac-Moody groups over fields in the sense that they have an RGD-system. This is a system of subgroups indexed by the roots of a root system and satisfying certain commutation relations.

Roughly speaking, there is a one-to-one correspondence between groups of Kac-Moody type and Moufang twin buildings. This correspondence was used in the last decade to prove several group theoretic results on RGD-systems and in particular on Kac-

Moody groups over fields.

In my talk I will explain RGD-systems and how they provide twin

buildings in a natural way. I will then present some of the group theoretic applications mentioned above and describe how twin buildings are used as a main tool in their proof.

Groups of Kac-Moody type are natural generalizations of Kac-Moody groups over fields in the sense that they have an RGD-system. This is a system of subgroups indexed by the roots of a root system and satisfying certain commutation relations.

Roughly speaking, there is a one-to-one correspondence between groups of Kac-Moody type and Moufang twin buildings. This correspondence was used in the last decade to prove several group theoretic results on RGD-systems and in particular on Kac-

Moody groups over fields.

In my talk I will explain RGD-systems and how they provide twin

buildings in a natural way. I will then present some of the group theoretic applications mentioned above and describe how twin buildings are used as a main tool in their proof.

### 2010/07/15

14:30-16:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Pieri rule and Pieri algebras for the orthogonal groups (ENGLISH)

**Soo Teck Lee**(Singapore National University)Pieri rule and Pieri algebras for the orthogonal groups (ENGLISH)

### 2010/06/08

17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Automorphism groups of causal Makarevich spaces (JAPANESE)

**Soji Kaneyuki**(Sophia University)Automorphism groups of causal Makarevich spaces (JAPANESE)

[ Abstract ]

Let G^ be a simple Lie group of Hermitian type and U^ be a maximal parabolic subgroup of G^ with abelian nilradical. The flag manifold M^= G^/ U^ is the Shilov

boundary of an irreducible bounded symmetric domain of tube type. M^ has the G-invariant causal structure. A causal Makarevich space is, by definition, an open symmetric G-orbit M in M^, endowed with the causal structure induced from that

of the ambient space M^, G being a reductive subgroup of G^. All symmetric cones fall in the class of causal Makarevich spaces.

In this talk, we determine the causal automorphism groups of all causal Makarevich spaces.

Let G^ be a simple Lie group of Hermitian type and U^ be a maximal parabolic subgroup of G^ with abelian nilradical. The flag manifold M^= G^/ U^ is the Shilov

boundary of an irreducible bounded symmetric domain of tube type. M^ has the G-invariant causal structure. A causal Makarevich space is, by definition, an open symmetric G-orbit M in M^, endowed with the causal structure induced from that

of the ambient space M^, G being a reductive subgroup of G^. All symmetric cones fall in the class of causal Makarevich spaces.

In this talk, we determine the causal automorphism groups of all causal Makarevich spaces.

### 2010/05/25

17:00-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On endoscopy, packets, and invariants (JAPANESE)

**Kaoru Hiraga**(Kyoto University)On endoscopy, packets, and invariants (JAPANESE)

[ Abstract ]

The theory of endoscopy came out of the Langlands functoriality and the trace formula.

In this talk, I will briefly explain what the endoscopy is, and talk about packet, formal degree and Whittaker normalization of transfer.

I would like to talk about the connection between these topics and the endoscopy.

The theory of endoscopy came out of the Langlands functoriality and the trace formula.

In this talk, I will briefly explain what the endoscopy is, and talk about packet, formal degree and Whittaker normalization of transfer.

I would like to talk about the connection between these topics and the endoscopy.

### 2010/05/18

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

On the eigenvalues of the Laplacian on locally symmetric hyperbolic spaces (ENGLISH)

**B. Speh**(Cornel University)On the eigenvalues of the Laplacian on locally symmetric hyperbolic spaces (ENGLISH)

[ Abstract ]

A famous Theorem of Selberg says that the non-zero eigenvalues of the Laplacian acting on functions on quotients of the upper half plane H by congruence subgroups of the integral modular group, are bounded away from zero, as the congruence subgroup varies. Analogous questions on Laplacians acting on differential forms of higher degree on locally symmetric spaces (functions may be thought of as differential forms of degree zero) have geometric implications for the cohomology of the locally symmetric space.

Let $X$ be the real hyperbolic n-space and $\\Gamma \\subset $ SO(n, 1) a congruence arithmetic subgroup. Bergeron conjectured that the eigenvalues of the Laplacian acting on the differential forms on $ X / \\Gamma $ are bounded from the below by a constant independent of the congruence subgroup. In the lecture I will show how one can use representation theory to show that this conjecture is true provided that it is true in the middle degree.

This is joint work with T.N. Venkataramana

A famous Theorem of Selberg says that the non-zero eigenvalues of the Laplacian acting on functions on quotients of the upper half plane H by congruence subgroups of the integral modular group, are bounded away from zero, as the congruence subgroup varies. Analogous questions on Laplacians acting on differential forms of higher degree on locally symmetric spaces (functions may be thought of as differential forms of degree zero) have geometric implications for the cohomology of the locally symmetric space.

Let $X$ be the real hyperbolic n-space and $\\Gamma \\subset $ SO(n, 1) a congruence arithmetic subgroup. Bergeron conjectured that the eigenvalues of the Laplacian acting on the differential forms on $ X / \\Gamma $ are bounded from the below by a constant independent of the congruence subgroup. In the lecture I will show how one can use representation theory to show that this conjecture is true provided that it is true in the middle degree.

This is joint work with T.N. Venkataramana

### 2010/05/11

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

On a finite $W$-algebra module structure on the space of

continuous Whittaker vectors for an irreducible Harish-Chandra module (ENGLISH)

**Hisayosi Matumoto**(the University of Tokyo)On a finite $W$-algebra module structure on the space of

continuous Whittaker vectors for an irreducible Harish-Chandra module (ENGLISH)

[ Abstract ]

Let $G$ be a real reductive Lie group. The space of continuous Whittaker vectors for an irreducible Harish-Chandra module has a structure of a module over a finite $W$-algebra. We have seen such modules are irreducible for groups of type A. However, there is a counterexample to the naive conjecture. We discuss a refined version of the conjecture and further examples in this talk.

Let $G$ be a real reductive Lie group. The space of continuous Whittaker vectors for an irreducible Harish-Chandra module has a structure of a module over a finite $W$-algebra. We have seen such modules are irreducible for groups of type A. However, there is a counterexample to the naive conjecture. We discuss a refined version of the conjecture and further examples in this talk.

### 2010/04/27

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

Restriction of Vogan-Zuckerman's derived functor modules to symmetric subgroups (JAPANESE)

**Yoshiki Oshima**(the University of Tokyo)Restriction of Vogan-Zuckerman's derived functor modules to symmetric subgroups (JAPANESE)

[ Abstract ]

We study the restriction of Vogan-Zuckerman derived functor modules $A_\\frak{q}(\\lambda)$ to symmetric subgroups.

An algebraic condition for the discrete decomposability of

$A_\\frak{q}(\\lambda)$ was given by Kobayashi, which offers a framework for the detailed study of branching law.

In this talk, when $A_\\frak{q}(\\lambda)$ is discretely decomposable,

we construct some of irreducible components occurring in the branching law and determine their associated variety.

We study the restriction of Vogan-Zuckerman derived functor modules $A_\\frak{q}(\\lambda)$ to symmetric subgroups.

An algebraic condition for the discrete decomposability of

$A_\\frak{q}(\\lambda)$ was given by Kobayashi, which offers a framework for the detailed study of branching law.

In this talk, when $A_\\frak{q}(\\lambda)$ is discretely decomposable,

we construct some of irreducible components occurring in the branching law and determine their associated variety.

### 2010/04/20

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

Proper actions of SL(2,R) on semisimple symmetric spaces (JAPANESE)

**Takayuki Okuda**(the University of Tokyo)Proper actions of SL(2,R) on semisimple symmetric spaces (JAPANESE)

[ Abstract ]

Complex irreducible symmetric spaces which admit proper SL(2,R)-actions were classified by Katsuki Teduka.

In this talk, we generalize Teduka's method and classify semisimple symmetric spaces which admit proper SL(2,R)-actions.

Complex irreducible symmetric spaces which admit proper SL(2,R)-actions were classified by Katsuki Teduka.

In this talk, we generalize Teduka's method and classify semisimple symmetric spaces which admit proper SL(2,R)-actions.

### 2010/04/15

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Generalized hypergeometric systems (ENGLISH)

**Uuganbayar Zunderiya**(Nagoya University)Generalized hypergeometric systems (ENGLISH)

[ Abstract ]

A new type of hypergeometric differential equations was introduced and studied by H. Sekiguchi. The investigated system of partial differential equation generalizes the Gauss-Aomoto-Gelfand system which in its turn stems from the classical set of differential relations for the solutions to a generic algebraic equation introduced by K.Mayr in 1937. Gauss-Aomoto-Gelfand systems can be expressed as the determinants of $2\\times 2$ matrices of derivations with respect to certain variables. H. Sekiguchi generalized this construction by looking at determinations of arbitrary $l\\times l$ matrices of derivations with respect to certain variables.

In this talk we study the dimension of global (and local) solutions to the generalized systems of Gauss-Aomoto-Gelfand hypergeometric systems. The main results in the talk are a combinatorial formula for the dimension of global (and local) solutions of the generalized Gauss-Aomoto-Gelfand system and a theorem on generic holonomicity of a certain class of such systems.

A new type of hypergeometric differential equations was introduced and studied by H. Sekiguchi. The investigated system of partial differential equation generalizes the Gauss-Aomoto-Gelfand system which in its turn stems from the classical set of differential relations for the solutions to a generic algebraic equation introduced by K.Mayr in 1937. Gauss-Aomoto-Gelfand systems can be expressed as the determinants of $2\\times 2$ matrices of derivations with respect to certain variables. H. Sekiguchi generalized this construction by looking at determinations of arbitrary $l\\times l$ matrices of derivations with respect to certain variables.

In this talk we study the dimension of global (and local) solutions to the generalized systems of Gauss-Aomoto-Gelfand hypergeometric systems. The main results in the talk are a combinatorial formula for the dimension of global (and local) solutions of the generalized Gauss-Aomoto-Gelfand system and a theorem on generic holonomicity of a certain class of such systems.

### 2010/04/06

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

On the characters of tempered modules of affine Hecke

algebras of classical type

**加藤周**(京都大学)On the characters of tempered modules of affine Hecke

algebras of classical type

[ Abstract ]

We present an inductive algorithm to compute the characters

of tempered modules of an affine Hecke algebras of classical

types, based on a new class of representations which we call

"tempered delimits". They have some geometric origin in the

eDL correspondence.

Our new algorithm has some advantage to the Lusztig-Shoji

algorithm (which also describes the characters of tempered

modules via generalized Green functions) in the sense it

enables us to tell how the characters of tempered modules

changes as the parameters vary.

This is a joint work with Dan Ciubotaru at Utah.

We present an inductive algorithm to compute the characters

of tempered modules of an affine Hecke algebras of classical

types, based on a new class of representations which we call

"tempered delimits". They have some geometric origin in the

eDL correspondence.

Our new algorithm has some advantage to the Lusztig-Shoji

algorithm (which also describes the characters of tempered

modules via generalized Green functions) in the sense it

enables us to tell how the characters of tempered modules

changes as the parameters vary.

This is a joint work with Dan Ciubotaru at Utah.

### 2010/02/19

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

Discrete groups acting on homogeneous spaces V

**Yves Benoist**(Orsay)Discrete groups acting on homogeneous spaces V

[ Abstract ]

I will focus on recent advances on our understanding of discrete subgroups of Lie groups.

I will first survey how ideas from semisimple algebraic groups, ergodic theory and representation theory help us to understand properties of these discrete subgroups.

I will then focus on a joint work with Jean-Francois Quint studying the dynamics of these discrete subgroups on finite volume homogeneous spaces and proving the following result:

We fix two integral matrices A and B of size d, of determinant 1, and such that no finite union of vector subspaces is invariant by A and B. We fix also an irrational point on the d-dimensional torus. We will then prove that for n large the set of images of this point by the words in A and B of length at most n becomes equidistributed in the torus.

I will focus on recent advances on our understanding of discrete subgroups of Lie groups.

I will first survey how ideas from semisimple algebraic groups, ergodic theory and representation theory help us to understand properties of these discrete subgroups.

I will then focus on a joint work with Jean-Francois Quint studying the dynamics of these discrete subgroups on finite volume homogeneous spaces and proving the following result:

We fix two integral matrices A and B of size d, of determinant 1, and such that no finite union of vector subspaces is invariant by A and B. We fix also an irrational point on the d-dimensional torus. We will then prove that for n large the set of images of this point by the words in A and B of length at most n becomes equidistributed in the torus.

### 2010/02/02

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Deformation of compact quotients of homogeneous spaces

**Fanny Kassel**(Orsay)Deformation of compact quotients of homogeneous spaces

[ Abstract ]

Let G/H be a reductive homogeneous space. In all known examples, if

G/H admits compact Clifford-Klein forms, then it admits "standard"

ones, by uniform lattices of some reductive subgroup L of G acting

properly on G/H. In order to obtain more generic Clifford-Klein forms,

we prove that for L of real rank 1, if one slightly deforms in G a

uniform lattice of L, then its action on G/H remains properly

discontinuous. As an application, we obtain compact quotients of SO(2,2n)/U(1,n) by Zariski-dense discrete subgroups of SO(2,2n) acting properly discontinuously.

Let G/H be a reductive homogeneous space. In all known examples, if

G/H admits compact Clifford-Klein forms, then it admits "standard"

ones, by uniform lattices of some reductive subgroup L of G acting

properly on G/H. In order to obtain more generic Clifford-Klein forms,

we prove that for L of real rank 1, if one slightly deforms in G a

uniform lattice of L, then its action on G/H remains properly

discontinuous. As an application, we obtain compact quotients of SO(2,2n)/U(1,n) by Zariski-dense discrete subgroups of SO(2,2n) acting properly discontinuously.

### 2010/01/12

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

代数的差分方程式の可解性と既約性

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20100112nishioka

**西岡斉治**(東京大学大学院数理科学研究科博士課程)代数的差分方程式の可解性と既約性

[ Abstract ]

差分代数の理論を使って,代数的差分方程式の代数函数解や超幾

何函数解の非存在や,存在する場合の特殊解の分類をする。

[ Reference URL ]差分代数の理論を使って,代数的差分方程式の代数函数解や超幾

何函数解の非存在や,存在する場合の特殊解の分類をする。

https://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20100112nishioka