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Lie群論・表現論セミナー
過去の記録 ～03/19｜次回の予定｜今後の予定 03/20～
| 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
2014年07月12日(土)
13:20-17:00 数理科学研究科棟(駒場) 126号室
Mikhail Kapranov 氏 (Kavli IPMU) 13:20-14:20
Perverse sheaves on hyperplane arrangements (ENGLISH)
Upper global nasis, cluster algebra and simplicity of tensor products of simple modules (ENGLISH)
Branching Problems of Representations of Real Reductive Groups (ENGLISH)
Mikhail Kapranov 氏 (Kavli IPMU) 13:20-14:20
Perverse sheaves on hyperplane arrangements (ENGLISH)
[ 講演概要 ]
Given an arrangement of hyperplanes in $R^n$, one has the complexified arrangement in $C^n$ and the corresponding category of perverse sheaves (smooth along the strata of the natural stratification).
The talk, based in a joint work with V. Schechtman, will present an explicit description of this category in terms of data associated to the face complex of the real arrangement. Such a description suggests a possibility of categorifying the concept of a oerverse sheaf in this and possibly in more general cases.
柏原正樹 氏 (京都大学数理解析研究所) 14:40-15:40Given an arrangement of hyperplanes in $R^n$, one has the complexified arrangement in $C^n$ and the corresponding category of perverse sheaves (smooth along the strata of the natural stratification).
The talk, based in a joint work with V. Schechtman, will present an explicit description of this category in terms of data associated to the face complex of the real arrangement. Such a description suggests a possibility of categorifying the concept of a oerverse sheaf in this and possibly in more general cases.
Upper global nasis, cluster algebra and simplicity of tensor products of simple modules (ENGLISH)
[ 講演概要 ]
One of the motivation of cluster algebras introduced by
Fomin and Zelevinsky is
multiplicative properties of upper global basis.
In this talk, I explain their relations, related conjectures by Besrnard Leclerc and the recent progress by the speaker with Seok-Jin Kang, Myungho Kima and Sejin Oh.
小林俊行 氏 (東京大学大学院数理科学研究科) 16:00-17:00One of the motivation of cluster algebras introduced by
Fomin and Zelevinsky is
multiplicative properties of upper global basis.
In this talk, I explain their relations, related conjectures by Besrnard Leclerc and the recent progress by the speaker with Seok-Jin Kang, Myungho Kima and Sejin Oh.
Branching Problems of Representations of Real Reductive Groups (ENGLISH)
[ 講演概要 ]
Branching problems ask how irreducible representations π of groups G "decompose" when restricted to subgroups G'.
For real reductive groups, branching problems include various important special cases, however, it is notorious that "infinite multiplicities" and "continuous spectra" may well happen in general even if (G,G') are natural pairs such as symmetric pairs.
By using analysis on (real) spherical varieties, we give a necessary and sufficient condition on the pair of reductive groups for the multiplicities to be always finite (and also to be of uniformly bounded). Further, we discuss "discretely decomposable restrictions" which allows us to apply algebraic tools in branching problems. Some classification results will be also presented.
If time permits, I will discuss some applications of branching laws of Zuckerman's derived functor modules to analysis on locally symmetric spaces with indefinite metric.
Branching problems ask how irreducible representations π of groups G "decompose" when restricted to subgroups G'.
For real reductive groups, branching problems include various important special cases, however, it is notorious that "infinite multiplicities" and "continuous spectra" may well happen in general even if (G,G') are natural pairs such as symmetric pairs.
By using analysis on (real) spherical varieties, we give a necessary and sufficient condition on the pair of reductive groups for the multiplicities to be always finite (and also to be of uniformly bounded). Further, we discuss "discretely decomposable restrictions" which allows us to apply algebraic tools in branching problems. Some classification results will be also presented.
If time permits, I will discuss some applications of branching laws of Zuckerman's derived functor modules to analysis on locally symmetric spaces with indefinite metric.
2014年07月12日(土)
09:30-11:45 数理科学研究科棟(駒場) 126号室
大島利雄 氏 (城西大学) 09:30-10:30
超幾何系とKac-Moodyルート系 (ENGLISH)
Representations of covering groups with multiplicity free K-types (ENGLISH)
大島利雄 氏 (城西大学) 09:30-10:30
超幾何系とKac-Moodyルート系 (ENGLISH)
[ 講演概要 ]
帯球関数やそれの一般化のHeckmann-Opdamの超幾何の解析のため,1次元
の特異集合への制限から常微分方程式の研究に興味を持った.
Fuchs型常微分方程式全体の空間にEuler変換などを通じてKac-Moodyルー
ト系のWeyl群が作用することが分かり,局所モノドロミーで決まらない
モジュライ空間の次元を不変量として,群軌道の有限性が明らかになった.
モジュライがないrigidな場合は自明な方程式に変換されるので具体的
解析が可能になり,逆にモジュライのある場合はPainleve方程式の構成
と分類への応用がある.これらは分岐のない不確定特異点も許す場合に
拡張されると共に,リジッドな場合は自然に多変数の超幾何への延長が
定義され,その解析に役立つ.古典的なAppellの超幾何などは後者に含
まれ,モノドロミーの可約性などがルート系の言葉で一般的に記述でき
る.これらの概説と共に,最近の結果や今後の問題ついて解説する.
Gordan Savin 氏 (the University of Utah) 10:45-11:45帯球関数やそれの一般化のHeckmann-Opdamの超幾何の解析のため,1次元
の特異集合への制限から常微分方程式の研究に興味を持った.
Fuchs型常微分方程式全体の空間にEuler変換などを通じてKac-Moodyルー
ト系のWeyl群が作用することが分かり,局所モノドロミーで決まらない
モジュライ空間の次元を不変量として,群軌道の有限性が明らかになった.
モジュライがないrigidな場合は自明な方程式に変換されるので具体的
解析が可能になり,逆にモジュライのある場合はPainleve方程式の構成
と分類への応用がある.これらは分岐のない不確定特異点も許す場合に
拡張されると共に,リジッドな場合は自然に多変数の超幾何への延長が
定義され,その解析に役立つ.古典的なAppellの超幾何などは後者に含
まれ,モノドロミーの可約性などがルート系の言葉で一般的に記述でき
る.これらの概説と共に,最近の結果や今後の問題ついて解説する.
Representations of covering groups with multiplicity free K-types (ENGLISH)
[ 講演概要 ]
Let g be a simple Lie algebra over complex numbers. McGovern has
described an ideal J in the enveloping algebra U such that U/J, considered as a g-module under the adjoint action, is a sum of all self-dual representations of g with multiplicity one. In a joint work with Loke, we prove that all (g,K)-modules annihilated by J have multiplicity free K-types, where K is defined by the Chevalley involution.
Let g be a simple Lie algebra over complex numbers. McGovern has
described an ideal J in the enveloping algebra U such that U/J, considered as a g-module under the adjoint action, is a sum of all self-dual representations of g with multiplicity one. In a joint work with Loke, we prove that all (g,K)-modules annihilated by J have multiplicity free K-types, where K is defined by the Chevalley involution.
2014年07月01日(火)
16:30-18:00 数理科学研究科棟(駒場) 126号室
Pablo Ramacher
氏 (Marburg University)
WONDERFUL VARIETIES. REGULARIZED TRACES AND CHARACTERS (ENGLISH)
Pablo Ramacher
氏 (Marburg University)
WONDERFUL VARIETIES. REGULARIZED TRACES AND CHARACTERS (ENGLISH)
[ 講演概要 ]
Let G be a connected reductive complex algebraic group with split real form $G^\\sigma$.
In this talk, we introduce a distribution character for the regular representation of $G^\\sigma$ on the real locus of a strict wonderful G-variety X, showing that on a certain open subset of $G^\\sigma$ of transversal elements it is locally integrable, and given by a sum over fixed points.
Let G be a connected reductive complex algebraic group with split real form $G^\\sigma$.
In this talk, we introduce a distribution character for the regular representation of $G^\\sigma$ on the real locus of a strict wonderful G-variety X, showing that on a certain open subset of $G^\\sigma$ of transversal elements it is locally integrable, and given by a sum over fixed points.
2014年06月17日(火)
16:30-18:00 数理科学研究科棟(駒場) 126号室
Pablo Ramacher 氏 (Marburg University)
SINGULAR EQUIVARIANT ASYMPTOTICS AND THE MOMENTUM MAP. RESIDUE FORMULAE IN EQUIVARIANT COHOMOLOGY (ENGLISH)
Pablo Ramacher 氏 (Marburg University)
SINGULAR EQUIVARIANT ASYMPTOTICS AND THE MOMENTUM MAP. RESIDUE FORMULAE IN EQUIVARIANT COHOMOLOGY (ENGLISH)
[ 講演概要 ]
Let M be a smooth manifold and G a compact connected Lie group acting on M by isometries. In this talk, we study the equivariant cohomology of the cotangent bundle of M, and relate it to the cohomology of the Marsden-Weinstein reduced space via certain residue formulae. In case of compact symplectic manifolds with a Hamiltonian G-action, similar residue formulae were derived by Jeffrey, Kirwan et al..
Let M be a smooth manifold and G a compact connected Lie group acting on M by isometries. In this talk, we study the equivariant cohomology of the cotangent bundle of M, and relate it to the cohomology of the Marsden-Weinstein reduced space via certain residue formulae. In case of compact symplectic manifolds with a Hamiltonian G-action, similar residue formulae were derived by Jeffrey, Kirwan et al..
2014年05月27日(火)
16:30-18:00 数理科学研究科棟(駒場) 126号室
渡部真樹 氏 (東京大学大学院数理科学研究科)
Schubert加群の構造とSchubert加群によるfiltrationについて (JAPANESE)
渡部真樹 氏 (東京大学大学院数理科学研究科)
Schubert加群の構造とSchubert加群によるfiltrationについて (JAPANESE)
[ 講演概要 ]
Schubert多項式を研究する道具の1つとして, KraskiewiczとPragaczによって 導入されたSchubert加群があります.
今回の発表では, Schubert加群の構造に関する新しい結果と, そこから得られる, 与えられた加群がSchubert加群によるfiltrationを持つ条件について話します.
また, この研究のもともとの動機はSchubert多項式に関するある問題を考えていたことなので, それについても話す予定です.
Schubert多項式を研究する道具の1つとして, KraskiewiczとPragaczによって 導入されたSchubert加群があります.
今回の発表では, Schubert加群の構造に関する新しい結果と, そこから得られる, 与えられた加群がSchubert加群によるfiltrationを持つ条件について話します.
また, この研究のもともとの動機はSchubert多項式に関するある問題を考えていたことなので, それについても話す予定です.
2014年05月13日(火)
16:30-18:00 数理科学研究科棟(駒場) 126号室
Ivan Cherednik 氏 (The University of North Carolina at Chapel Hill, RIMS
)
Global q,t-hypergeometric and q-Whittaker functions (ENGLISH)
Ivan Cherednik 氏 (The University of North Carolina at Chapel Hill, RIMS
)
Global q,t-hypergeometric and q-Whittaker functions (ENGLISH)
[ 講演概要 ]
The lectures will be devoted to the new theory of global
difference hypergeometric and Whittaker functions, one of
the major applications of the double affine Hecke algebras
and a breakthrough in the classical harmonic analysis. They
integrate the Ruijsenaars-Macdonald difference QMBP and
"Q-Toda" (any root systems), and are analytic everywhere
("global") with superb asymptotic behavior.
The definition of the global functions was suggested about
14 years ago; it is conceptually different from the definition
Heine gave in 1846, which remained unchanged and unchallenged
since then. Algebraically, the new functions are closer to
Bessel functions than to the classical hypergeometric and
Whittaker functions. The analytic theory of these functions was
completed only recently (the speaker and Jasper Stokman).
The construction is based on DAHA. The global functions are defined
as reproducing kernels of Fourier-DAHA transforms. Their
specializations are Macdonald polynomials, which is a powerful
generalization of the Shintani and Casselman-Shalika p-adic formulas.
If time permits, the connection of the Harish-Chandra theory of global
q-Whittaker functions will be discussed with the Givental-Lee formula
(Gromov-Witten invariants of flag varieties) and its generalizations due
to Braverman and Finkelberg (algebraic theory of affine flag varieties).
The lectures will be devoted to the new theory of global
difference hypergeometric and Whittaker functions, one of
the major applications of the double affine Hecke algebras
and a breakthrough in the classical harmonic analysis. They
integrate the Ruijsenaars-Macdonald difference QMBP and
"Q-Toda" (any root systems), and are analytic everywhere
("global") with superb asymptotic behavior.
The definition of the global functions was suggested about
14 years ago; it is conceptually different from the definition
Heine gave in 1846, which remained unchanged and unchallenged
since then. Algebraically, the new functions are closer to
Bessel functions than to the classical hypergeometric and
Whittaker functions. The analytic theory of these functions was
completed only recently (the speaker and Jasper Stokman).
The construction is based on DAHA. The global functions are defined
as reproducing kernels of Fourier-DAHA transforms. Their
specializations are Macdonald polynomials, which is a powerful
generalization of the Shintani and Casselman-Shalika p-adic formulas.
If time permits, the connection of the Harish-Chandra theory of global
q-Whittaker functions will be discussed with the Givental-Lee formula
(Gromov-Witten invariants of flag varieties) and its generalizations due
to Braverman and Finkelberg (algebraic theory of affine flag varieties).
2014年04月15日(火)
16:30-18:00 数理科学研究科棟(駒場) 126号室
土岡俊介 氏 (東京大学大学院数理科学研究科)
Toward the graded Cartan invariants of the symmetric groups (JAPANESE)
土岡俊介 氏 (東京大学大学院数理科学研究科)
Toward the graded Cartan invariants of the symmetric groups (JAPANESE)
[ 講演概要 ]
We propose a graded analog of Hill's conjecture which is equivalent to K\\"ulshammer-Olsson-Robinson's conjecture on the generalized Cartan invariants of the symmetric groups.
We give justifications for it and discuss implications between the variants.
Some materials are based on the joint work with Anton Evseev.
We propose a graded analog of Hill's conjecture which is equivalent to K\\"ulshammer-Olsson-Robinson's conjecture on the generalized Cartan invariants of the symmetric groups.
We give justifications for it and discuss implications between the variants.
Some materials are based on the joint work with Anton Evseev.
2014年01月14日(火)
16:30-18:00 数理科学研究科棟(駒場) 126号室
森真樹 氏 (東京大学大学院数理科学研究科)
セルラー代数の手法によるHecke-Cliffordスーパー代数の単純加群の分類
(JAPANESE)
森真樹 氏 (東京大学大学院数理科学研究科)
セルラー代数の手法によるHecke-Cliffordスーパー代数の単純加群の分類
(JAPANESE)
[ 講演概要 ]
Hecke--Cliffordスーパー代数はA型岩堀--Hecke代数のスーパー版である。
その単純加群の分類は、Brundan, Kleshchevと土岡により
アフィンLie代数の圏論化の手法を用いて行われた。しかしこの構成は
とても抽象的であり実際に単純加群の構造を詳しく調べるのは難しい。
そこで本講演では、より具体的な単純加群の構成方法を紹介する。
これはGrahamとLehrerによるセルラー代数の理論を拡張した手法である。
ここではSpecht加群のスーパー類似にCliffordスーパー代数が
右から作用することが鍵となる。森田コンテクストと呼ばれる
道具を用いることで、このCliffordスーパー代数の単純加群から
Hecke--Cliffordスーパー代数の単純加群を作ることができる。
Hecke--Cliffordスーパー代数はA型岩堀--Hecke代数のスーパー版である。
その単純加群の分類は、Brundan, Kleshchevと土岡により
アフィンLie代数の圏論化の手法を用いて行われた。しかしこの構成は
とても抽象的であり実際に単純加群の構造を詳しく調べるのは難しい。
そこで本講演では、より具体的な単純加群の構成方法を紹介する。
これはGrahamとLehrerによるセルラー代数の理論を拡張した手法である。
ここではSpecht加群のスーパー類似にCliffordスーパー代数が
右から作用することが鍵となる。森田コンテクストと呼ばれる
道具を用いることで、このCliffordスーパー代数の単純加群から
Hecke--Cliffordスーパー代数の単純加群を作ることができる。
2013年12月17日(火)
16:30-17:30 数理科学研究科棟(駒場) 126号室
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)
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
Simon Gindikin 氏 (Rutgers University (USA))
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
Ronald King 氏 (the University of Southampton)
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
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)
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
Vaibhav Vaish 氏 (the Institute of Mathematical Sciences) 14:30-15:20
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) 14:30-15:20
Weightless 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.
Yuichiro Tanaka 氏 (the University of Tokyo) 15:40-16:10Using 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.
Visible 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.
Pampa Paul 氏 (Indian Statistical Institute, Kolkata) 16:10-16:40The 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.
Holomorphic 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.
Dipendra Prasad 氏 (Tata Institute of Fundamental Research) 16:50-17:40Let $G_0$ 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.
Branching 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
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)
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
Benjamin Harris 氏 (Louisiana State University (USA))
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
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)
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
Michael Pevzner 氏 (Reims University) 16:30-17:30
Non-standard models for small representations of GL(n,R) (ENGLISH)
Degenerate principal series of symplectic groups (ENGLISH)
Michael Pevzner 氏 (Reims University) 16:30-17:30
Non-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.
Pierre Clare 氏 (Penn. State University, USA) 17:30-18:30We 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.
Degenerate 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
笹木集夢 氏 (東海大学)
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
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)
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
Nizar Demni 氏 (Université de Rennes 1)
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
Nizar Demni 氏 (Université de Rennes 1)
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
Simon Goodwin 氏 (Birmingham University)
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
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)
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
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)
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
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)
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.
