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Lie Groups and Representation Theory
Seminar information archive ～04/19｜Next seminar｜Future seminars 04/20～
| Date, time & place | Tuesday 16:30 - 18:00 126Room #126 (Graduate School of Math. Sci. Bldg.) |
|---|
Seminar information archive
2014/05/13
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Ivan Cherednik (The University of North Carolina at Chapel Hill, RIMS
)
Global q,t-hypergeometric and q-Whittaker functions (ENGLISH)
Ivan Cherednik (The University of North Carolina at Chapel Hill, RIMS
)
Global q,t-hypergeometric and q-Whittaker functions (ENGLISH)
[ Abstract ]
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).
2014/04/15
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Shunsuke Tsuchioka (the University of Tokyo)
Toward the graded Cartan invariants of the symmetric groups (JAPANESE)
Shunsuke Tsuchioka (the University of Tokyo)
Toward the graded Cartan invariants of the symmetric groups (JAPANESE)
[ Abstract ]
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.
2014/01/14
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Masaki Mori (the University of Tokyo)
A cellular classification of simple modules of the Hecke-Clifford
superalgebra (JAPANESE)
Masaki Mori (the University of Tokyo)
A cellular classification of simple modules of the Hecke-Clifford
superalgebra (JAPANESE)
[ Abstract ]
The Hecke--Clifford superalgebra is a super version of
the Iwahori--Hecke algebra of type A. Its simple modules
are classified by Brundan, Kleshchev and Tsuchioka using
a method of categorification of affine Lie algebras.
However their constructions are too abstract to study in practice.
In this talk, we introduce a more concrete way to produce its
simple modules with a generalized theory of cellular algebras
which is originally developed by Graham and Lehrer.
In our construction the key is that there is a right action of
the Clifford superalgebra on the super-analogue of the Specht module.
With the help of the notion of the Morita context, a simple module
of the Hecke--Clifford superalgebra is made from that of
the Clifford superalgebra.
The Hecke--Clifford superalgebra is a super version of
the Iwahori--Hecke algebra of type A. Its simple modules
are classified by Brundan, Kleshchev and Tsuchioka using
a method of categorification of affine Lie algebras.
However their constructions are too abstract to study in practice.
In this talk, we introduce a more concrete way to produce its
simple modules with a generalized theory of cellular algebras
which is originally developed by Graham and Lehrer.
In our construction the key is that there is a right action of
the Clifford superalgebra on the super-analogue of the Specht module.
With the help of the notion of the Morita context, a simple module
of the Hecke--Clifford superalgebra is made from that of
the Clifford superalgebra.
2013/12/17
16:30-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Koichi Kaizuka (University of Tsukuba)
A characterization of the $L^{2}$-range of the
Poisson transform on symmetric spaces of noncompact type (JAPANESE)
Koichi Kaizuka (University of Tsukuba)
A characterization of the $L^{2}$-range of the
Poisson transform on symmetric spaces of noncompact type (JAPANESE)
[ Abstract ]
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.
2013/11/19
16:30-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Simon Gindikin (Rutgers University (USA))
Horospheres, wonderfull compactification and c-function (JAPANESE)
Simon Gindikin (Rutgers University (USA))
Horospheres, wonderfull compactification and c-function (JAPANESE)
[ Abstract ]
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
2013/11/11
16:30-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Ronald King (the University of Southampton)
Alternating sign matrices, primed shifted tableaux and Tokuyama
factorisation theorems (ENGLISH)
Ronald King (the University of Southampton)
Alternating sign matrices, primed shifted tableaux and Tokuyama
factorisation theorems (ENGLISH)
[ Abstract ]
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.
2013/11/07
13:30-14:20 Room #000 (Graduate School of Math. Sci. Bldg.)
Toshiyuki Kobayashi (the University of Tokyo)
Global Geometry and Analysis on Locally Pseudo-Riemannian Homogeneous Spaces (ENGLISH)
Toshiyuki Kobayashi (the University of Tokyo)
Global Geometry and Analysis on Locally Pseudo-Riemannian Homogeneous Spaces (ENGLISH)
[ Abstract ]
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.
2013/11/07
14:30-17:40 Room #000 (Graduate School of Math. Sci. Bldg.)
Vaibhav Vaish (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 (the Institute of Mathematical Sciences) 14:30-15:20
Weightless cohomology of algebraic varieties (ENGLISH)
[ Abstract ]
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 (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)
[ Abstract ]
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 (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)
[ Abstract ]
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 (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)
[ Abstract ]
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.
2013/10/29
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Yuichiro Tanaka (the University of Tokyo, Graduate School of Mathematical Sciences)
Geometry of multiplicity-free representations of SO(N) and visible actions (JAPANESE)
Yuichiro Tanaka (the University of Tokyo, Graduate School of Mathematical Sciences)
Geometry of multiplicity-free representations of SO(N) and visible actions (JAPANESE)
[ Abstract ]
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.
2013/10/22
17:00-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Benjamin Harris (Louisiana State University (USA))
Representation Theory and Microlocal Analysis (ENGLISH)
Benjamin Harris (Louisiana State University (USA))
Representation Theory and Microlocal Analysis (ENGLISH)
[ Abstract ]
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.
2013/04/30
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Hisayosi MATUMOTO (Graduate School of Mathematical Sciences, the University of Tokyo)
The homomorphisms between scalar generalized Verma modules of gl(n,C) with regular infinitesimal characters (JAPANESE)
Hisayosi MATUMOTO (Graduate School of Mathematical Sciences, the University of Tokyo)
The homomorphisms between scalar generalized Verma modules of gl(n,C) with regular infinitesimal characters (JAPANESE)
[ Abstract ]
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.
2013/04/16
16:30-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Michael Pevzner (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 (Reims University) 16:30-17:30
Non-standard models for small representations of GL(n,R) (ENGLISH)
[ Abstract ]
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 (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)
[ Abstract ]
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.
2013/04/09
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Atsumu Sasaki (Tokai University)
A characterization of non-tube type Hermitian symmetric spaces by visible actions
(JAPANESE)
Atsumu Sasaki (Tokai University)
A characterization of non-tube type Hermitian symmetric spaces by visible actions
(JAPANESE)
[ Abstract ]
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.
2013/04/02
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Yoshiki Oshima (Kavli IPMU, the University of Tokyo)
Discrete branching laws of Zuckerman's derived functor modules (JAPANESE)
Yoshiki Oshima (Kavli IPMU, the University of Tokyo)
Discrete branching laws of Zuckerman's derived functor modules (JAPANESE)
[ Abstract ]
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.
2013/02/05
17:30-19:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Nizar Demni (Université de Rennes 1)
Dunkl processes assciated with dihedral systems, II (ENGLISH)
Nizar Demni (Université de Rennes 1)
Dunkl processes assciated with dihedral systems, II (ENGLISH)
[ Abstract ]
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.
2013/02/04
17:30-19:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Nizar Demni (Université de Rennes 1)
Dunkl processes assciated with dihedral systems, I (ENGLISH)
Nizar Demni (Université de Rennes 1)
Dunkl processes assciated with dihedral systems, I (ENGLISH)
[ Abstract ]
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.
2013/01/22
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Simon Goodwin (Birmingham University)
Representation theory of finite W-algebras (ENGLISH)
Simon Goodwin (Birmingham University)
Representation theory of finite W-algebras (ENGLISH)
[ Abstract ]
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).
2013/01/08
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Soji Kaneyuki (Sophia University )
On the group of holomorphic and anti-holomorphic transformations
of a compact Hermitian symmetric space and the $G$-structure (JAPANESE)
Soji Kaneyuki (Sophia University )
On the group of holomorphic and anti-holomorphic transformations
of a compact Hermitian symmetric space and the $G$-structure (JAPANESE)
[ Abstract ]
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.
2012/12/11
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Tatsuyuki Hikita (Kyoto University)
Affine Springer fibers of type A and combinatorics of diagonal
coinvariants
Affine Springer fibers of type A and combinatorics of diagonal
coinvariants (JAPANESE)
Tatsuyuki Hikita (Kyoto University)
Affine Springer fibers of type A and combinatorics of diagonal
coinvariants
Affine Springer fibers of type A and combinatorics of diagonal
coinvariants (JAPANESE)
[ Abstract ]
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.
2012/11/29
16:30-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)
Masaki Watanabe (the University of Tokyo)
On a relation between certain character values of symmetric groups (JAPANESE)
Masaki Watanabe (the University of Tokyo)
On a relation between certain character values of symmetric groups (JAPANESE)
[ Abstract ]
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.
2012/11/27
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Hiroshi Konno (the University of Tokyo)
Convergence of Kahler to real polarizations on flag manifolds (JAPANESE)
Hiroshi Konno (the University of Tokyo)
Convergence of Kahler to real polarizations on flag manifolds (JAPANESE)
[ Abstract ]
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.
2012/11/20
16:30-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Ali Baklouti (Sfax University)
On the geometry of discontinuous subgroups acting on some homogeneous spaces (ENGLISH)
Ali Baklouti (Sfax University)
On the geometry of discontinuous subgroups acting on some homogeneous spaces (ENGLISH)
[ Abstract ]
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.
2012/11/13
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Oskar Hamlet (Chalmers University)
Tight maps, a classification (ENGLISH)
Oskar Hamlet (Chalmers University)
Tight maps, a classification (ENGLISH)
[ Abstract ]
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.
2012/11/06
16:30-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Takayuki Okuda (the University of Tokyo)
An explicit construction of spherical designs on S^3 (JAPANESE)
Takayuki Okuda (the University of Tokyo)
An explicit construction of spherical designs on S^3 (JAPANESE)
[ Abstract ]
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.
2012/07/24
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Toshihisa Kubo (the University of Tokyo)
The Dynkin index and conformally invariant systems of Heisenberg parabolic type (ENGLISH)
Toshihisa Kubo (the University of Tokyo)
The Dynkin index and conformally invariant systems of Heisenberg parabolic type (ENGLISH)
[ Abstract ]
Recently, Barchini-Kable-Zierau systematically constructed conformally invariant systems of differential operators using Heisenberg parabolic subalgebras. When they built such systems, two constants, which are defined as the constant of proportionality between two expressions,played an important role. In this talk we give concrete and uniform expressions for these constants. To do so the Dynkin index of a finite dimensional representation of a complex simple Lie algebra plays a key role.
Recently, Barchini-Kable-Zierau systematically constructed conformally invariant systems of differential operators using Heisenberg parabolic subalgebras. When they built such systems, two constants, which are defined as the constant of proportionality between two expressions,played an important role. In this talk we give concrete and uniform expressions for these constants. To do so the Dynkin index of a finite dimensional representation of a complex simple Lie algebra plays a key role.
