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Lie Groups and Representation Theory
Seminar information archive ～05/07｜Next seminar｜Future seminars 05/08～
| Date, time & place | Tuesday 16:30 - 18:00 126Room #126 (Graduate School of Math. Sci. Bldg.) |
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Seminar information archive
2021/10/26
17:00-18:00 Room #online (Graduate School of Math. Sci. Bldg.)
Masatoshi KITAGAWA (Waseda University)
Applications of uniform bounded families of g-modules to branching problems (Japanese)
Masatoshi KITAGAWA (Waseda University)
Applications of uniform bounded families of g-modules to branching problems (Japanese)
[ Abstract ]
Using the notion of uniformly bounded families of g-modules introduced in arXiv:2109.05556, we can prove several finiteness and uniform boundedness results of multiplicities in branching laws and induced representations.
After the introduction of such results, I will explain how to obtain the necessary and sufficient condition for the uniform boundedness of multiplicities in branching laws given in arXiv:2109.05555.
Using the notion of uniformly bounded families of g-modules introduced in arXiv:2109.05556, we can prove several finiteness and uniform boundedness results of multiplicities in branching laws and induced representations.
After the introduction of such results, I will explain how to obtain the necessary and sufficient condition for the uniform boundedness of multiplicities in branching laws given in arXiv:2109.05555.
2021/10/19
17:00-18:00 Room #online (Graduate School of Math. Sci. Bldg.)
Hiroyoshi Tamori (Hokkaido University)
Classification of type A analogues of minimal representations
(Japanese)
Hiroyoshi Tamori (Hokkaido University)
Classification of type A analogues of minimal representations
(Japanese)
[ Abstract ]
If $\mathfrak{g}$ is a simple Lie algebra not of type A, the enveloping algebra $U(\mathfrak{g})$ has a unique completely prime primitive ideal whose associated variety equals the closure of the minimal nilpotent orbit. The ideal is called the Joseph Ideal. An irreducible admissible representation of a simple Lie group is called minimal if the annihilator of the underlying $(\mathfrak{g},\mathfrak{k})$-modules is given by the Joseph ideal. Minimal representations are known to have simple $\mathfrak{k}$-type decompositions (called pencil), and a simple Lie group has at most two minimal representations up to complex conjugate. In this talk, we consider the type A analogues for the above statements.
If $\mathfrak{g}$ is a simple Lie algebra not of type A, the enveloping algebra $U(\mathfrak{g})$ has a unique completely prime primitive ideal whose associated variety equals the closure of the minimal nilpotent orbit. The ideal is called the Joseph Ideal. An irreducible admissible representation of a simple Lie group is called minimal if the annihilator of the underlying $(\mathfrak{g},\mathfrak{k})$-modules is given by the Joseph ideal. Minimal representations are known to have simple $\mathfrak{k}$-type decompositions (called pencil), and a simple Lie group has at most two minimal representations up to complex conjugate. In this talk, we consider the type A analogues for the above statements.
2021/10/05
17:00-18:00 Room #Online (Graduate School of Math. Sci. Bldg.)
Toshiyuki KOBAYASHI (The University of Tokyo)
Bounded multiplicity in the branching problems of "small" infinite-dimensional representations (Japanese)
Toshiyuki KOBAYASHI (The University of Tokyo)
Bounded multiplicity in the branching problems of "small" infinite-dimensional representations (Japanese)
[ Abstract ]
I plan to explain a geometric criterion for the bounded multiplicity property of “small” infinite-dimensional
representations of real reductive Lie groups in branching problems.
Applying the criterion to symmetric pairs, we give a full description of the triples H ⊂ G ⊃ G' such that any irreducible admissible representations of G with H-distinguished vectors have the bounded multiplicity property when restricted to the subgroup G'.
The precise results are available in [Adv. Math. 2021, Section 7] and arXiv:2109.14424, and I plan to give some flavor.
I plan to explain a geometric criterion for the bounded multiplicity property of “small” infinite-dimensional
representations of real reductive Lie groups in branching problems.
Applying the criterion to symmetric pairs, we give a full description of the triples H ⊂ G ⊃ G' such that any irreducible admissible representations of G with H-distinguished vectors have the bounded multiplicity property when restricted to the subgroup G'.
The precise results are available in [Adv. Math. 2021, Section 7] and arXiv:2109.14424, and I plan to give some flavor.
2021/07/28
17:00-18:00 Room #Online (Graduate School of Math. Sci. Bldg.)
Yoshiki Oshima (Osaka University, Graduate School of Information Science and Technology)
Collapsing Ricci-flat metrics and a priori estimate for the Monge-Ampere equation
(Japanese)
Yoshiki Oshima (Osaka University, Graduate School of Information Science and Technology)
Collapsing Ricci-flat metrics and a priori estimate for the Monge-Ampere equation
(Japanese)
[ Abstract ]
Yau proved the Calabi conjecture by using a priori estimate for the Monge-Ampere equation. Recently, for a Calabi-Yau manifold with a fiber space structure, the behavior of Ricci-flat metrics collapsing to a Kahler class of the base space was studied by Gross-Tosatti-Zhang, etc. The Gromov-Hausdorff convergence of K3 surfaces to spheres obtained by a joint work with Yuji Odaka (arXiv:1810.07685) is also based on those estimates for solutions to the Monge-Ampere equation. In this talk, I would like to discuss how an estimate of solutions to differential equations deduces the existence of canonical metrics and the Gromov-
Hausdorff convergence.
Yau proved the Calabi conjecture by using a priori estimate for the Monge-Ampere equation. Recently, for a Calabi-Yau manifold with a fiber space structure, the behavior of Ricci-flat metrics collapsing to a Kahler class of the base space was studied by Gross-Tosatti-Zhang, etc. The Gromov-Hausdorff convergence of K3 surfaces to spheres obtained by a joint work with Yuji Odaka (arXiv:1810.07685) is also based on those estimates for solutions to the Monge-Ampere equation. In this talk, I would like to discuss how an estimate of solutions to differential equations deduces the existence of canonical metrics and the Gromov-
Hausdorff convergence.
2021/07/20
17:00-18:00 Room #Online (Graduate School of Math. Sci. Bldg.)
Hiroyoshi Tamori (Hokkaido University)
On the existence of a nonzero linear period (Japanese)
Hiroyoshi Tamori (Hokkaido University)
On the existence of a nonzero linear period (Japanese)
[ Abstract ]
Let $(G,H)$ be a symmetric pair $(\mathrm{GL}(n,\mathbb{H}),\mathrm{GL}(n,\mathbb{C}))$ or $(\mathrm{GL}(2n,\mathbb{R}),\mathrm{GL}(n,\mathbb{C}))$. It was proved by Broussous-Matringe that for an irreducible smooth admissible Fr\'{e}chet representation $\pi$ of $G$ of moderate growth, the dimension of the space of $H$-linear period of $\pi$ is not greater then one. We give some necessary condition for the existence of a nonzero $H$-linear period of $\pi$, which proves the archimedean case of a conjecture by Prasad and Takloo-Bighash. Our approach is based on the $H$-orbit decomposition of the flag variety of $G$, and homology of principal series representations. This is a joint work with Miyu Suzuki (Kanazawa University).
Let $(G,H)$ be a symmetric pair $(\mathrm{GL}(n,\mathbb{H}),\mathrm{GL}(n,\mathbb{C}))$ or $(\mathrm{GL}(2n,\mathbb{R}),\mathrm{GL}(n,\mathbb{C}))$. It was proved by Broussous-Matringe that for an irreducible smooth admissible Fr\'{e}chet representation $\pi$ of $G$ of moderate growth, the dimension of the space of $H$-linear period of $\pi$ is not greater then one. We give some necessary condition for the existence of a nonzero $H$-linear period of $\pi$, which proves the archimedean case of a conjecture by Prasad and Takloo-Bighash. Our approach is based on the $H$-orbit decomposition of the flag variety of $G$, and homology of principal series representations. This is a joint work with Miyu Suzuki (Kanazawa University).
2021/07/13
17:00-18:00 Room #Online (Graduate School of Math. Sci. Bldg.)
Yoshiki Oshima (Osaka University, Graduate School of Information Science and Technology)
Compactification of locally symmetric spaces and collapsing of canonical Kahler metrics (Japanese)
Yoshiki Oshima (Osaka University, Graduate School of Information Science and Technology)
Compactification of locally symmetric spaces and collapsing of canonical Kahler metrics (Japanese)
[ Abstract ]
The moduli spaces of Abelian varieties and K3 surfaces are known to have a structure of locally symmetric spaces. Around 1960, a finite number of compactifications of locally symmetric spaces are constructed by Ichiro Satake. In this talk, based on a joint work with Yuji Odaka (arXiv:1810:07685), we will see that one of Satake compactifications parametrizes limits of canonical (Ricci-flat) Kahler metrics on Abelian varieties and K3 surfaces.
The moduli spaces of Abelian varieties and K3 surfaces are known to have a structure of locally symmetric spaces. Around 1960, a finite number of compactifications of locally symmetric spaces are constructed by Ichiro Satake. In this talk, based on a joint work with Yuji Odaka (arXiv:1810:07685), we will see that one of Satake compactifications parametrizes limits of canonical (Ricci-flat) Kahler metrics on Abelian varieties and K3 surfaces.
2021/07/06
17:00-18:00 Room #Online (Graduate School of Math. Sci. Bldg.)
Taito Tauchi (Kyushu University )
A counterexample to a Q-series analogue of Casselman's subrepresentation theorem (Japanese)
Taito Tauchi (Kyushu University )
A counterexample to a Q-series analogue of Casselman's subrepresentation theorem (Japanese)
[ Abstract ]
Let G be a real reductive Lie group, Q a parabolic subgroup of G, and π an irreducible admissible representation of G. We say that π belongs to Q-series if it occurs as a subquotient of some degenerate principal series representation induced from Q. Then, any irreducible admissible representation belongs to P-series by Harish-Chandra’s subquotient theorem, where P is a minimal parabolic subgroup of G. On the other hand, Casselman’s subrepresentation theorem implies any representation belonging to P-series can be realized as a
subrepresentation of some principal series representation induced from P. In this talk, we discuss a counterexample to a Q-series analogue of this subrepresentation theorem. More precisely, we show that there exists an irreducible admissible representation belonging to Q-series, which can not be realized as a subrepresentation of any degenerate
principal series representation induced from Q.
Let G be a real reductive Lie group, Q a parabolic subgroup of G, and π an irreducible admissible representation of G. We say that π belongs to Q-series if it occurs as a subquotient of some degenerate principal series representation induced from Q. Then, any irreducible admissible representation belongs to P-series by Harish-Chandra’s subquotient theorem, where P is a minimal parabolic subgroup of G. On the other hand, Casselman’s subrepresentation theorem implies any representation belonging to P-series can be realized as a
subrepresentation of some principal series representation induced from P. In this talk, we discuss a counterexample to a Q-series analogue of this subrepresentation theorem. More precisely, we show that there exists an irreducible admissible representation belonging to Q-series, which can not be realized as a subrepresentation of any degenerate
principal series representation induced from Q.
2021/06/29
17:00-18:00 Room #Online (Graduate School of Math. Sci. Bldg.)
Hidenori Fujiwara (Kindai University)
Polynomial conjectures for nilpotent Lie groups
(Japanese)
Hidenori Fujiwara (Kindai University)
Polynomial conjectures for nilpotent Lie groups
(Japanese)
[ Abstract ]
Let G = exp g be a connected and simply connected nilpotent Lie group with Lie algebra g. Let H = exp h be an analytic subgroup of G with Lie algebra h and χ a unitary character of H. We consider the monomial representation τ = ind^G_H χ of G. It is well known that the multiplicities in the irreducible disintegration of τ are either uniformly bounded or uniformly equal to ∞. In the former case, we say that τ has finite multiplicities.
Now let D_τ (G/H) be the algebra of the G-invariant differential operators on the fiber bundle over G/H associated to the data (H,χ). This algebra is commutative if and only if τ has finite multiplicities. In
1992 Corwin-Greenleaf presented the following polynomial conjecture :
when τ has finite multiplicities, the algebra D_τ (G/H) is isomorphic to the algebra C[Γ_τ]^H of the H-invariant polynomial functions on the affine subspace Γ_τ = {l ∈ g^* ; l |_h = - √ -1 dχ} of g^* .
It is well known in the representation theory of groups that between the two operations of induction and restriction there is a kind of duality. So, we think about a polynomial conjecture for restrictions. Let G be as above a connected and simply connected nilpotent Lie group and π an irreducible unitary representation of G. Let K be an analytic subgroup of G, and we consider the restriction π|_K of π to K. This time also it is known that the multiplicities in the irreducible disintegration of π|_K are either uniformly bounded or uniformly equal to ∞. In the former case, we say that π|_K has finite multiplicities and we assume
this eventuality. Let U(g) be the enveloping algebra of g_C, and we consider the algebra (U(g)/kerπ)_K of invariant differential operators. This means the set of the K-invariant elements. This algebra is commutative if and only if π|_K has finite multiplicities. In this case, is the algebra (U(g)/kerπ)^K isomorphic to the algebra C[Ω(π)]^K of the K-invariant polynomial functions on Ω(π)? Here, Ω(π) denotes the coadjoint orbit of G corresponding to π.
We would like to prove these two polynomial conjectures.
Let G = exp g be a connected and simply connected nilpotent Lie group with Lie algebra g. Let H = exp h be an analytic subgroup of G with Lie algebra h and χ a unitary character of H. We consider the monomial representation τ = ind^G_H χ of G. It is well known that the multiplicities in the irreducible disintegration of τ are either uniformly bounded or uniformly equal to ∞. In the former case, we say that τ has finite multiplicities.
Now let D_τ (G/H) be the algebra of the G-invariant differential operators on the fiber bundle over G/H associated to the data (H,χ). This algebra is commutative if and only if τ has finite multiplicities. In
1992 Corwin-Greenleaf presented the following polynomial conjecture :
when τ has finite multiplicities, the algebra D_τ (G/H) is isomorphic to the algebra C[Γ_τ]^H of the H-invariant polynomial functions on the affine subspace Γ_τ = {l ∈ g^* ; l |_h = - √ -1 dχ} of g^* .
It is well known in the representation theory of groups that between the two operations of induction and restriction there is a kind of duality. So, we think about a polynomial conjecture for restrictions. Let G be as above a connected and simply connected nilpotent Lie group and π an irreducible unitary representation of G. Let K be an analytic subgroup of G, and we consider the restriction π|_K of π to K. This time also it is known that the multiplicities in the irreducible disintegration of π|_K are either uniformly bounded or uniformly equal to ∞. In the former case, we say that π|_K has finite multiplicities and we assume
this eventuality. Let U(g) be the enveloping algebra of g_C, and we consider the algebra (U(g)/kerπ)_K of invariant differential operators. This means the set of the K-invariant elements. This algebra is commutative if and only if π|_K has finite multiplicities. In this case, is the algebra (U(g)/kerπ)^K isomorphic to the algebra C[Ω(π)]^K of the K-invariant polynomial functions on Ω(π)? Here, Ω(π) denotes the coadjoint orbit of G corresponding to π.
We would like to prove these two polynomial conjectures.
2021/06/22
17:00-18:00 Room #Online (Graduate School of Math. Sci. Bldg.)
Yosuke MORITA (Kyoto University)
Cartan projections of some nonreductive subgroups and the existence problem of compact Clifford-Klein forms (Japanese)
Yosuke MORITA (Kyoto University)
Cartan projections of some nonreductive subgroups and the existence problem of compact Clifford-Klein forms (Japanese)
[ Abstract ]
Let G be a reductive Lie group, H a closed subgroup of G, and Γ a discrete subgroup of G.
According to the Kobayashi-Benoist criterion, the properness of the Γ-action on G/H is determined by the Cartan projections of H and Γ.
Although the Cartan projections of nonreductive subgroups are usually difficult to compute, there are some exceptions. Using them, we give some examples of homogeneous spaces of reductive type that do not admit compact Clifford-Klein forms.
Let G be a reductive Lie group, H a closed subgroup of G, and Γ a discrete subgroup of G.
According to the Kobayashi-Benoist criterion, the properness of the Γ-action on G/H is determined by the Cartan projections of H and Γ.
Although the Cartan projections of nonreductive subgroups are usually difficult to compute, there are some exceptions. Using them, we give some examples of homogeneous spaces of reductive type that do not admit compact Clifford-Klein forms.
2021/06/15
17:00-18:00 Room #Online (Graduate School of Math. Sci. Bldg.)
Toshiyuki KOBAYASHI (The University of Tokyo)
Limit algebras and tempered representations (Japanese)
Toshiyuki KOBAYASHI (The University of Tokyo)
Limit algebras and tempered representations (Japanese)
[ Abstract ]
I plan to discuss the new connection between the following four (apparently unrelated) topics:
1. (analysis) Tempered unitary representations on homogeneous spaces
2. (combinatorics) Convex polyhedral cones
3. (topology) Limit algebras
4. (symplectic geometry) Quantization of coadjoint orbits
based on a series of joint papers with Y. Benoist "Tempered homogeneous spaces I-IV".
I plan to discuss the new connection between the following four (apparently unrelated) topics:
1. (analysis) Tempered unitary representations on homogeneous spaces
2. (combinatorics) Convex polyhedral cones
3. (topology) Limit algebras
4. (symplectic geometry) Quantization of coadjoint orbits
based on a series of joint papers with Y. Benoist "Tempered homogeneous spaces I-IV".
2021/06/08
17:00-18:00 Room #Online (Graduate School of Math. Sci. Bldg.)
Kazuki KANNAKA (RIKEN iTHEMS)
The multiplicities of stable eigenvalues on compact anti-de Sitter 3-manifolds (Japanese)
Kazuki KANNAKA (RIKEN iTHEMS)
The multiplicities of stable eigenvalues on compact anti-de Sitter 3-manifolds (Japanese)
[ Abstract ]
A \textit{pseudo-Riemannian locally symmetric space} is the quotient manifold $\Gamma\backslash G/H$ of a semisimple symmetric space $G/H$ by a discontinuous group $\Gamma$.
Toshiyuki Kobayashi initiated the study of spectral analysis of \textit{intrinsic differential operators} (such as the Laplacian) of a pseudo-Rimannian locally symmetric space. Unlike the classical Riemannian setting,
the Laplacian of a pseudo-Rimannian locally symmetric space is no longer an elliptic differential operator.
In its spectral analysis, new phenomena different from those in the Riemannian setting have been discovered in recent years, following pioneering works by Kassel-Kobayashi.
For instance, they studied the behavior of eigenvalues of intrinsic differential operators of $\Gamma\backslash G/H$ when deforming a discontinuous group $\Gamma$. As a special case, they found infinitely many \textit{stable
eigenvalues} of the (hyperbolic) Laplacian of a compact anti-de Sitter $3$-manifold $\Gamma\backslash
\mathrm{SO}(2,2)/\mathrm{SO}(2,1)$ ([Adv.\ Math.\ 2016]).
In this talk, I would like to explain recent results about the \textit{multiplicities} of stable eigenvalues in the anti-de Sitter setting.
A \textit{pseudo-Riemannian locally symmetric space} is the quotient manifold $\Gamma\backslash G/H$ of a semisimple symmetric space $G/H$ by a discontinuous group $\Gamma$.
Toshiyuki Kobayashi initiated the study of spectral analysis of \textit{intrinsic differential operators} (such as the Laplacian) of a pseudo-Rimannian locally symmetric space. Unlike the classical Riemannian setting,
the Laplacian of a pseudo-Rimannian locally symmetric space is no longer an elliptic differential operator.
In its spectral analysis, new phenomena different from those in the Riemannian setting have been discovered in recent years, following pioneering works by Kassel-Kobayashi.
For instance, they studied the behavior of eigenvalues of intrinsic differential operators of $\Gamma\backslash G/H$ when deforming a discontinuous group $\Gamma$. As a special case, they found infinitely many \textit{stable
eigenvalues} of the (hyperbolic) Laplacian of a compact anti-de Sitter $3$-manifold $\Gamma\backslash
\mathrm{SO}(2,2)/\mathrm{SO}(2,1)$ ([Adv.\ Math.\ 2016]).
In this talk, I would like to explain recent results about the \textit{multiplicities} of stable eigenvalues in the anti-de Sitter setting.
2021/06/01
17:30-18:30 Room #Online (Graduate School of Math. Sci. Bldg.)
Joint with Tuesday Seminar on Topology. Online.
Masatoshi KITAGAWA (Waseda University)
On the discrete decomposability and invariants of representations of real reductive Lie groups (Japanese)
Joint with Tuesday Seminar on Topology. Online.
Masatoshi KITAGAWA (Waseda University)
On the discrete decomposability and invariants of representations of real reductive Lie groups (Japanese)
[ Abstract ]
A problem to determine the behavior of the restriction of an irreducible group representation to a subgroup is called the branching problem. The restriction of an irreducible representation is not irreducible in general, and if the representation is unitary, the restriction has an irreducible decomposition described by a direct integral. The decomposition can be regarded as a generalization of spectral decomposition of unitary operators, and has continuous spectrum and discrete spectrum in general. If the decomposition has no continuous spectrum, the representation is said to be discretely decomposable.
Discretely decomposable branching laws are technically easy to deal with, and in the setting, it is relatively easy to extract information about representations of a small subgroup from that of a large group. The following applications are known. It is known that the operators, called the Rankin--Cohen brackets which make a new automorphic form from a automorphic form, can be obtained as
intertwiner from discretely decomposable representations to irreducible representations. Many generalizations of the operators are obtained recently. The discrete decomposability is used to construct discrete spectrum of the space of L^2 functions on homogeneous spaces (T. Kobayashi, J.
Funct. Anal. ('98)).
In this talk, I will give several criterion about the discrete decomposability and G'-admissibility based on the general theory and criterion given by T. Kobayashi (Invent. math. '94, Annals of Math. '98, Invent. math. '98).
The criterion are written by associated varieties (algebraic invariants), wave front sets (analytic invariants) and topological structure of representations.
A problem to determine the behavior of the restriction of an irreducible group representation to a subgroup is called the branching problem. The restriction of an irreducible representation is not irreducible in general, and if the representation is unitary, the restriction has an irreducible decomposition described by a direct integral. The decomposition can be regarded as a generalization of spectral decomposition of unitary operators, and has continuous spectrum and discrete spectrum in general. If the decomposition has no continuous spectrum, the representation is said to be discretely decomposable.
Discretely decomposable branching laws are technically easy to deal with, and in the setting, it is relatively easy to extract information about representations of a small subgroup from that of a large group. The following applications are known. It is known that the operators, called the Rankin--Cohen brackets which make a new automorphic form from a automorphic form, can be obtained as
intertwiner from discretely decomposable representations to irreducible representations. Many generalizations of the operators are obtained recently. The discrete decomposability is used to construct discrete spectrum of the space of L^2 functions on homogeneous spaces (T. Kobayashi, J.
Funct. Anal. ('98)).
In this talk, I will give several criterion about the discrete decomposability and G'-admissibility based on the general theory and criterion given by T. Kobayashi (Invent. math. '94, Annals of Math. '98, Invent. math. '98).
The criterion are written by associated varieties (algebraic invariants), wave front sets (analytic invariants) and topological structure of representations.
2021/05/18
17:00-17:30 Room #Online (Graduate School of Math. Sci. Bldg.)
Mamoru UEDA (Kyoto University)
Affine Yangians and rectangular W-algebras (Japanese)
Mamoru UEDA (Kyoto University)
Affine Yangians and rectangular W-algebras (Japanese)
[ Abstract ]
Motivated by the generalized AGT conjecture, in this talk I will construct surjective homomorphisms from Guay's affine Yangians to the universal enveloping algebras of rectangular W-algebras of type A.
This result is a super affine analogue of a result of Ragoucy and Sorba, which gave surjective homomorphisms from finite Yangians of type A to rectangular finite W-algebras of type A.
Motivated by the generalized AGT conjecture, in this talk I will construct surjective homomorphisms from Guay's affine Yangians to the universal enveloping algebras of rectangular W-algebras of type A.
This result is a super affine analogue of a result of Ragoucy and Sorba, which gave surjective homomorphisms from finite Yangians of type A to rectangular finite W-algebras of type A.
2021/05/11
17:00-18:00 Room #Online (Graduate School of Math. Sci. Bldg.)
Online
Ryosuke NAKAHAMA (Kyushu University)
Computation of weighted Bergman inner products on bounded symmetric domains and restriction to subgroups (Japanese)
Online
Ryosuke NAKAHAMA (Kyushu University)
Computation of weighted Bergman inner products on bounded symmetric domains and restriction to subgroups (Japanese)
[ Abstract ]
Let $D¥subset M(r,¥mathbb{C})$ be the bounded symmetric domain, and we consider the weighted Bergman space $¥mathcal{H}_¥lambda(D)$ on $D$. Then $SU(r,r)$ acts unitarily on $¥mathcal{H}_¥lambda(D)$.
In this seminar, we compute explicitly the inner products for some polynomials on $¥operatorname{Alt}(r,¥mathbb{C})$, $¥operatorname{Sym}(r,¥mathbb{C})¥subset M(r,¥mathbb{C})$, and prove that the inner products are given by multivariate hypergeometric polynomials when the polynomials are some powers of the determinants or the Pfaffians.
As an application, we present the results on the construction of symmetry breaking operators from $SU(r,r)$ to $Sp(r,¥mathbb{R})$ or $SO^*(2r)$.
Let $D¥subset M(r,¥mathbb{C})$ be the bounded symmetric domain, and we consider the weighted Bergman space $¥mathcal{H}_¥lambda(D)$ on $D$. Then $SU(r,r)$ acts unitarily on $¥mathcal{H}_¥lambda(D)$.
In this seminar, we compute explicitly the inner products for some polynomials on $¥operatorname{Alt}(r,¥mathbb{C})$, $¥operatorname{Sym}(r,¥mathbb{C})¥subset M(r,¥mathbb{C})$, and prove that the inner products are given by multivariate hypergeometric polynomials when the polynomials are some powers of the determinants or the Pfaffians.
As an application, we present the results on the construction of symmetry breaking operators from $SU(r,r)$ to $Sp(r,¥mathbb{R})$ or $SO^*(2r)$.
2020/07/14
17:30-18:30 Room ## (Graduate School of Math. Sci. Bldg.)
Joint with Tuesday Seminar on Topology. Online.
Takayuki Okuda (Hiroshima University)
Kobayashi's properness criterion and totally geodesic submanifolds in locally symmetric spaces (Japanese)
Joint with Tuesday Seminar on Topology. Online.
Takayuki Okuda (Hiroshima University)
Kobayashi's properness criterion and totally geodesic submanifolds in locally symmetric spaces (Japanese)
[ Abstract ]
Let G be a Lie group and X a homogeneous G-space.
A discrete subgroup of G acting on X properly is called a discontinuous group for X.
We are interested in constructions and classifications of discontinuous groups for a given X.
It is well-known that if the isotropies of G on X are compact, any closed subgroup acts on X properly.
However, the cases where the isotropies are non-compact, the same claim does not hold in general.
Let us consider the case where G is a linear reductive.
In this situation, T. Kobayashi [Math. Ann. (1989)], [J. Lie Theory (1996)] gave a criterion for the properness of the action on a homogeneous G-space X of closed subgroups in G.
In this talk, we consider homogeneous G-spaces of reductive types realized as families of totally geodesic submanifolds in non-compact Riemannian symmetric spaces.
As a main result, we give a translation of Kobayashi's criterion within the framework of Riemannian geometry.
In particular, for a torsion-free discrete subgroup of G, the criterion can be stated in terms of totally geodesic submanifolds in the Riemannian locally symmetric space corresponding to the subgroup in G.
Let G be a Lie group and X a homogeneous G-space.
A discrete subgroup of G acting on X properly is called a discontinuous group for X.
We are interested in constructions and classifications of discontinuous groups for a given X.
It is well-known that if the isotropies of G on X are compact, any closed subgroup acts on X properly.
However, the cases where the isotropies are non-compact, the same claim does not hold in general.
Let us consider the case where G is a linear reductive.
In this situation, T. Kobayashi [Math. Ann. (1989)], [J. Lie Theory (1996)] gave a criterion for the properness of the action on a homogeneous G-space X of closed subgroups in G.
In this talk, we consider homogeneous G-spaces of reductive types realized as families of totally geodesic submanifolds in non-compact Riemannian symmetric spaces.
As a main result, we give a translation of Kobayashi's criterion within the framework of Riemannian geometry.
In particular, for a torsion-free discrete subgroup of G, the criterion can be stated in terms of totally geodesic submanifolds in the Riemannian locally symmetric space corresponding to the subgroup in G.
2020/02/05
15:00-16:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Simon Gindikin (Rutgers University)
Direct inversion of the horospherical transform on Riemannian symmetric spaces (English)
Simon Gindikin (Rutgers University)
Direct inversion of the horospherical transform on Riemannian symmetric spaces (English)
[ Abstract ]
It was a problem of Gelfand to find an inversion of the horospherical transform directly and as a result to find directly the Plancherel formula.
I will give such an inversion and it gives a formula different from Harish-Chandra's one.
It was a problem of Gelfand to find an inversion of the horospherical transform directly and as a result to find directly the Plancherel formula.
I will give such an inversion and it gives a formula different from Harish-Chandra's one.
2020/01/28
10:00-16:40 Room # (Graduate School of Math. Sci. Bldg.)
Taito Tauchi (The University of Tokyo) 10:00-11:00
Relationship between orbit decomposition on the flag varieties and multiplicities of induced representations (English)
Mikhail Kapranov (Kavli IPMU) 11:20-12:20
TBA (English)
Michael Pevzner (University of Reims) 14:00-15:00
From Symmetry breaking toward holographic transform in representation theory (English)
Leticia Barchini (Oklahoma University) 15:40-16:40
Cells of Harish-Chandra modules
(English)
Taito Tauchi (The University of Tokyo) 10:00-11:00
Relationship between orbit decomposition on the flag varieties and multiplicities of induced representations (English)
Mikhail Kapranov (Kavli IPMU) 11:20-12:20
TBA (English)
Michael Pevzner (University of Reims) 14:00-15:00
From Symmetry breaking toward holographic transform in representation theory (English)
Leticia Barchini (Oklahoma University) 15:40-16:40
Cells of Harish-Chandra modules
(English)
2020/01/27
9:30-16:30 Room # (Graduate School of Math. Sci. Bldg.)
Joseph Bernstein (Tel Aviv and The University of Tokyo) 10:00-11:00
TBA (English)
Toshiyuki Kobayashi (The University of Tokyo) 11:20-12:20
Regular Representations on Homogeneous Spaces (English)
Laura Geatti (University of Roma) 14:00-15:00
The adapted hyper-K\"ahler structure on the tangent bundle of a Hermitian symmetric space (English)
Simon Gindikin (Rutgers University) 15:30-16:30
UNIVERSAL NATURE OF THE HOROSPHERICAL TRANSFORM IN SYMMETRIC SPACES (English)
Joseph Bernstein (Tel Aviv and The University of Tokyo) 10:00-11:00
TBA (English)
Toshiyuki Kobayashi (The University of Tokyo) 11:20-12:20
Regular Representations on Homogeneous Spaces (English)
Laura Geatti (University of Roma) 14:00-15:00
The adapted hyper-K\"ahler structure on the tangent bundle of a Hermitian symmetric space (English)
Simon Gindikin (Rutgers University) 15:30-16:30
UNIVERSAL NATURE OF THE HOROSPHERICAL TRANSFORM IN SYMMETRIC SPACES (English)
2020/01/21
14:00-16:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Joseph Bernstein (Tel Aviv University)
On Plancherel measure (English)
Joseph Bernstein (Tel Aviv University)
On Plancherel measure (English)
2019/10/30
16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Quentin Labriet (Reims University)
On holographic transform (English)
Quentin Labriet (Reims University)
On holographic transform (English)
[ Abstract ]
In representation theory, decomposing the restriction of a given representation $¥pi$ of a Lie group $G$ to an appropriate subgroup $G'$ is an important issue referred to as a branching law. In this context, one can define symmetry breaking operators, as $G'$-intertwining operators between the restriction $¥pi¥vert_{G'}$ and its irreducible
components. Going in the opposite direction gives rise to holographic operators and the notion of holographic transform.
I will illustrate this construction by two examples :
- the diagonal case where one considers the restriction problem for $¥pi$ being an outer product of two holomorphic discrete series representations, $G=SL(2,R)¥times SL(2,R)$ and $G'=SL(2,R)$.
- the conformal case for the restriction of a scalar valued holomorphic discrete series representation $¥pi$ of $G=SO(2,n)$ to $G'=SO(2,n-1)$.
I will then explain different methods for an explicit construction of such holographic operators in these cases, and present some of my results and open problems in this direction.
In representation theory, decomposing the restriction of a given representation $¥pi$ of a Lie group $G$ to an appropriate subgroup $G'$ is an important issue referred to as a branching law. In this context, one can define symmetry breaking operators, as $G'$-intertwining operators between the restriction $¥pi¥vert_{G'}$ and its irreducible
components. Going in the opposite direction gives rise to holographic operators and the notion of holographic transform.
I will illustrate this construction by two examples :
- the diagonal case where one considers the restriction problem for $¥pi$ being an outer product of two holomorphic discrete series representations, $G=SL(2,R)¥times SL(2,R)$ and $G'=SL(2,R)$.
- the conformal case for the restriction of a scalar valued holomorphic discrete series representation $¥pi$ of $G=SO(2,n)$ to $G'=SO(2,n-1)$.
I will then explain different methods for an explicit construction of such holographic operators in these cases, and present some of my results and open problems in this direction.
2019/10/23
16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Clemens Weiske (Aarhus University)
Symmetry breaking and unitary branching laws for finite-multiplicity pairs of rank one (English)
Clemens Weiske (Aarhus University)
Symmetry breaking and unitary branching laws for finite-multiplicity pairs of rank one (English)
[ Abstract ]
Let (G,G’) be a real reductive finite multiplicity pair of rank one, i.e. a rank one real reductive group G with reductive subgroup G’, such that the space of symmetry breaking operators (SBOs) between all (smooth admissible) irreducible representations is finite dimensional.
We give a classification of SBOs between spherical principal series representations of G and G’, essentially generalizing the results on (O(1,n+1),O(1,n)) of Kobayashi—Speh (2015). Moreover we show how to decompose unitary representations occurring in (not necessarily) spherical principal series representations of G in terms of unitary G’ representations, by making use of the knowledge gathered in the classification of the SBOs and the structure of the open P’orbit in G/P as a homogenous G’-space, where P’ is a minimal parabolic in G’ and P is a minimal parabolic in G. This includes the construction of discrete spectra in the restriction of complementary series representations and unitarizable composition factors.
Let (G,G’) be a real reductive finite multiplicity pair of rank one, i.e. a rank one real reductive group G with reductive subgroup G’, such that the space of symmetry breaking operators (SBOs) between all (smooth admissible) irreducible representations is finite dimensional.
We give a classification of SBOs between spherical principal series representations of G and G’, essentially generalizing the results on (O(1,n+1),O(1,n)) of Kobayashi—Speh (2015). Moreover we show how to decompose unitary representations occurring in (not necessarily) spherical principal series representations of G in terms of unitary G’ representations, by making use of the knowledge gathered in the classification of the SBOs and the structure of the open P’orbit in G/P as a homogenous G’-space, where P’ is a minimal parabolic in G’ and P is a minimal parabolic in G. This includes the construction of discrete spectra in the restriction of complementary series representations and unitarizable composition factors.
2018/12/11
17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)
Motoki Takigiku (the University of Tokyo)
A Pieri-type formula and a factorization formula for K-k-Schur functions
Motoki Takigiku (the University of Tokyo)
A Pieri-type formula and a factorization formula for K-k-Schur functions
[ Abstract ]
We give a Pieri-type formula for the sum of K-k-Schur functions \sum_{\mu\le\lambda}g^{(k)}_{\mu} over a principal order ideal of the poset of k-bounded partitions under the strong Bruhat order, which sum we denote by \widetilde{g}^{(k)}_{\lambda}. As an application of this, we also give a k-rectangle factorization formula \widetilde{g}^{(k)}_{R_t\cup\lambda}=\widetilde{g}^{(k)}_{R_t} \widetilde{g}^{(k)}_{\lambda}
where R_t=(t^{k+1-t}), analogous to that of k-Schur functions s^{(k)}_{R_t\cup \lambda}=s^{(k)}_{R_t}s^{(k)}_{\lambda}.
We give a Pieri-type formula for the sum of K-k-Schur functions \sum_{\mu\le\lambda}g^{(k)}_{\mu} over a principal order ideal of the poset of k-bounded partitions under the strong Bruhat order, which sum we denote by \widetilde{g}^{(k)}_{\lambda}. As an application of this, we also give a k-rectangle factorization formula \widetilde{g}^{(k)}_{R_t\cup\lambda}=\widetilde{g}^{(k)}_{R_t} \widetilde{g}^{(k)}_{\lambda}
where R_t=(t^{k+1-t}), analogous to that of k-Schur functions s^{(k)}_{R_t\cup \lambda}=s^{(k)}_{R_t}s^{(k)}_{\lambda}.
2018/12/03
17:00-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Ali Baklouti (Faculté des Sciences de Sfax)
Monomial representations of discrete type and differential operators. (English)
Ali Baklouti (Faculté des Sciences de Sfax)
Monomial representations of discrete type and differential operators. (English)
[ Abstract ]
Let $G$ be an exponential solvable Lie group and $\tau$ a monomial representation of $G$, an induced representation from a connected closed subgroup of $G$ of a unitary character. It is well known that $\tau$ disintegrates into irreducible factors and the multiplicities of each isotypic component are explicitly determined. In the case where $G$ is nilpotent, these multiplicities are either finite or infinite almost everywhere, with respect to the disintegration's measure. We associate to $\tau$ an algebra of differential operators and it is shown that in the nilpotent case, the commutativity of this algebra is equivalent to the finiteness of the multiplicities of $\tau$. In the exponential case, we define the notion of monomial representation of discrete type. In this case, we show that such an equivalence does not hold and this answers a question posed by M. Duflo. This is a joint work with H. Fujiwara and J. Ludwig.
Let $G$ be an exponential solvable Lie group and $\tau$ a monomial representation of $G$, an induced representation from a connected closed subgroup of $G$ of a unitary character. It is well known that $\tau$ disintegrates into irreducible factors and the multiplicities of each isotypic component are explicitly determined. In the case where $G$ is nilpotent, these multiplicities are either finite or infinite almost everywhere, with respect to the disintegration's measure. We associate to $\tau$ an algebra of differential operators and it is shown that in the nilpotent case, the commutativity of this algebra is equivalent to the finiteness of the multiplicities of $\tau$. In the exponential case, we define the notion of monomial representation of discrete type. In this case, we show that such an equivalence does not hold and this answers a question posed by M. Duflo. This is a joint work with H. Fujiwara and J. Ludwig.
2018/03/12
15:00-16:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Christian Ikenmeyer (Max-Planck-Institut fur Informatik)
Plethysms and Kronecker coefficients in geometric complexity theory
Christian Ikenmeyer (Max-Planck-Institut fur Informatik)
Plethysms and Kronecker coefficients in geometric complexity theory
[ Abstract ]
Research on Kronecker coefficients and plethysms gained significant momentum when the topics were connected with geometric complexity theory, an approach towards computational complexity lower bounds via algebraic geometry and representation theory. This talk is about several recent results that were obtained with geometric complexity theory as motivation, namely the NP-hardness of deciding the positivity of Kronecker coefficients and an inequality between rectangular Kronecker coefficients and plethysm coefficients. While the proof of the former statement is mainly combinatorial, the proof of the latter statement interestingly uses insights from algebraic complexity theory. As far as we know algebraic complexity theory has never been used before to prove an inequality between representation theoretic multiplicities.
Research on Kronecker coefficients and plethysms gained significant momentum when the topics were connected with geometric complexity theory, an approach towards computational complexity lower bounds via algebraic geometry and representation theory. This talk is about several recent results that were obtained with geometric complexity theory as motivation, namely the NP-hardness of deciding the positivity of Kronecker coefficients and an inequality between rectangular Kronecker coefficients and plethysm coefficients. While the proof of the former statement is mainly combinatorial, the proof of the latter statement interestingly uses insights from algebraic complexity theory. As far as we know algebraic complexity theory has never been used before to prove an inequality between representation theoretic multiplicities.
2017/10/24
17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Reiko Miyaoka (Tohoku University)
Approach from the submanifold theory to the FLoer homology of Lagrangian intersections (JAPANESE)
Reiko Miyaoka (Tohoku University)
Approach from the submanifold theory to the FLoer homology of Lagrangian intersections (JAPANESE)
[ Abstract ]
The Gauss map images of isoparametric hypersurfaces in the spheres supply a rich family of minimal Lagrangian submanifolds of the complex hyperquadric Q_n(C). In simple cases, these are real forms of Q_n(C), and their Floer homology is known. In this talk, we consider the case when the number of distinct principal curvatures is 3,4,6, and report our results, which do not directly follow from FOOO’s theory. This is a joint work with Hiroshi Iriyeh (Ibaraki U.), Hui Ma (Tsinghua U.) and Yoshihiro Ohnita (Osaka City U.).
The Gauss map images of isoparametric hypersurfaces in the spheres supply a rich family of minimal Lagrangian submanifolds of the complex hyperquadric Q_n(C). In simple cases, these are real forms of Q_n(C), and their Floer homology is known. In this talk, we consider the case when the number of distinct principal curvatures is 3,4,6, and report our results, which do not directly follow from FOOO’s theory. This is a joint work with Hiroshi Iriyeh (Ibaraki U.), Hui Ma (Tsinghua U.) and Yoshihiro Ohnita (Osaka City U.).
