## Lie Groups and Representation Theory

Date, time & place Tuesday 16:30 - 18:00 126Room #126 (Graduate School of Math. Sci. Bldg.)

Seminar information archive

### 2022/06/28

17:00-18:00   Room #online (Graduate School of Math. Sci. Bldg.)
Ryosuke Nakahama (Kyushu University)
Computation of weighted Bergman inner products on bounded symmetric domains and Plancherel-type formulas for $(Sp(2r,\mathbb{R}),Sp(r,\mathbb{R})\times Sp(r,\mathbb{R}))$ (Japanese)
[ Abstract ]
Let $(G,G_1)=(G,(G^\sigma)_0)$ be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces $D_1=G_1/K_1\subset D=G/K$, realized as bounded symmetric domains in complex vector spaces $\mathfrak{p}^+_1:=(\mathfrak{p}^+)^\sigma\subset\mathfrak{p}^+$ respectively. Then the universal covering group $\widetilde{G}$ of $G$ acts unitarily on the weighted Bergman space $\mathcal{H}_\lambda(D)\subset\mathcal{O}(D)=\mathcal{O}_\lambda(D)$ on $D$ for sufficiently large $\lambda$. Its restriction to the subgroup $\widetilde{G}_1$ decomposes discretely and multiplicity-freely, and its branching law is given explicitly by Hua--Kostant--Schmid--Kobayashi's formula in terms of the $\widetilde{K}_1$-decomposition of the space $\mathcal{P}(\mathfrak{p}^+_2)$ of polynomials on $\mathfrak{p}^+_2:=(\mathfrak{p}^+)^{-\sigma}\subset\mathfrak{p}^+$. Our goal is to understand the decomposition of the restriction $\mathcal{H}_\lambda(D)|_{\widetilde{G}_1}$ by studying the weighted Bergman inner product on each $\widetilde{K}_1$-type in $\mathcal{P}(\mathfrak{p}^+_2)\subset\mathcal{H}_\lambda(D)$.
Today we mainly deal with the symmetric pair $(G,G_1)=(Sp(2r,\mathbb{R}),Sp(r,\mathbb{R})\times Sp(r,\mathbb{R}))$.

### 2022/05/17

17:00-18:00   Room #on line (Graduate School of Math. Sci. Bldg.)
Takashi Satomi (The University of Tokyo)
Estimate of the optimal constant of convolution inequalities on unimodular
locally compact groups
(Japanese)
[ Abstract ]
Some convolution inequalities (Young's inequality, the reverse Young's inequality, the Hausdorff--Young inequality) known for a long time on $\mathbb{R}$ can be generalized for any unimodular locally compact group.
In this seminar, we estimate the optimal constants (the ratio of both sides such that these inequalities are optimal) of these inequalities from above and below, and discuss that these estimates are the best for $G=\mathbb{R}$.

### 2022/05/10

17:00-18:00   Room #on line (Graduate School of Math. Sci. Bldg.)
Koichi Arashi (Nagoya University)
Holomorphic multiplier representations over bounded homogeneous domains (Japanese)
[ Abstract ]
I will talk about unitarizations in the spaces of holomorphic sections of equivariant holomorphic line bundles over bounded homogeneous domains. We consider the identity components of algebraic groups acting transitively on the domains. The main part of this talk is a classification of such unitary representations.
We discuss an explicit description of the classification for a specific five-dimensional non-symmetric bounded homogeneous domain to illustrate the method of the classification (K. Arashi, "Holomorphic multiplier representations for bounded homogeneous domains", Journal of Lie Theory 30, 1091-1116 (2020)).

### 2022/04/26

17:00-18:00   Room #online (Graduate School of Math. Sci. Bldg.)
Joint with Tuesday Seminar on Topology
Yoshiki Oshima (The University of Tokyo)
On the existence of discrete series for homogeneous spaces (Japanese)
[ Abstract ]
When a Lie group $G$ acts transitively on a manifold $X$, an irreducible subrepresentation of $L^2(X)$ is called a discrete series representation of $X$. One may ask which homogeneous space $X$ has a discrete series representation. For reductive symmetric spaces, it is known that the existence of discrete series is equivalent to a rank condition by works of Flensted-Jensen, T.Matsuki, and T.Oshima. The problem for general reductive homogeneous spaces was considered by T.Kobayashi and a sufficient condition for the existence of discrete series was obtained by using his theory of admissible restriction. In this talk, we would like to
see another sufficient condition for general homogeneous spaces and also the case of their line bundles in terms of the orbit method.

### 2022/04/19

17:30-18:30   Room #online (Graduate School of Math. Sci. Bldg.)
Joint with Tuesday Seminar on Topology
Toshihisa Kubo (Ryukoku University)
On the classification and construction of conformal symmetry breaking operators for anti-de Sitter spaces
(Japanese)
[ Abstract ]
Let $X$ be a smooth manifold and $Y$ a smooth submanifold of $X$. Take $G' \subset G$ to be a pair of Lie groups that act on $Y \subset X$, respectively. Consider a $G'$-intertwining differential operator $\mathcal{D}$ from the space of smooth sections for a $G$-equivariant vector bundle over $X$ to that for a $G'$-equivariant vector bundle over $Y$. Toshiyuki Kobayashi called such a differential operator $\mathcal{D}$ a \emph{differential symmetry breaking operator} (differential SBO for short)
([T. Kobayashi, Differential Geom. Appl. (2014)]).

In [Kobayashi-K-Pevzner, Lecture Notes in Math. 2170 (2016)], we explicitly constructed and classified all the differential SBOs from the space of differential $i$-forms $\mathcal{E}^i(S^n)$ over the standard
Riemann sphere $S^n$ to that of differential $j$-forms $\mathcal{E}^j(S^{n-1})$ over the totally geodesic hypersphere $S^{n-1}$.
In this talk, by extending the results in a Riemannian setting, we discuss about the classification and construction of differential SBOs in a pseudo-Riemannian setting such as anti-de Sitter spaces and hyperbolic spaces. This is a joint work with Toshiyuki Kobayashi and Michael Pevzner.

### 2022/04/05

17:00-17:30   Room #online (Graduate School of Math. Sci. Bldg.)
Toshiyuki KOBAYASHI (The University of Tokyo)
Note on the restriction of minimal representations with respect to reductive symmetric pairs (Japanese)
[ Abstract ]
I discuss briefly some abstract feature of branching problems with focus on the restriction of minimal representations with respect to reductive symmetric pairs.

### 2022/04/05

17:30-18:30   Room #online (Graduate School of Math. Sci. Bldg.)
Junko INOUE (Tottori University)
Estimate of the norm of the $L^p$-Fourier transform on compact extensions of locally compact groups
(Japanese)
[ Abstract ]
The classical Hausdorff-Young theorem for locally compact abelian groups is generalized by Kunze for unimodular locally compact groups.
When the group $G$ is of type I, the abstract Plancherel theorem gives a decomposition of the regular representation into a direct integral of irreducible representations through the Fourier transform;
By the Hausdorff-Young theorem generalized by Kunze, for exponents $p$ $(1 < p \leq 2)$ and ${p'}=p/(p-1)$, the Fourier transform yields a bounded operator $\mathcal{F}^p:L^p(G)\to L^{p'}(\widehat{G})$, where $L^{p'}(\widehat{G})$ is the $L^{p'}$ space of measurable fields of the Schatten class operators on the unitary dual $\widehat{G}$ of $G$.
Under this setting, we are concerned with the norm $\|\mathcal{F}^p(G)\|$ of the $L^p$-Fourier transform $\mathcal{F}^p$.

Let $G$ be a separable unimodular locally compact group of type I,and $N$ be a type I, unimodular, closed normal subgroup of $G$. Suppose $G/N$ is compact. Then we show the inequality $\|\mathcal{F}^p(G)\|\leq\|\mathcal F^p(N)\|$ for $1< p \leq 2$.
This result is a joint work with Ali Baklouti
(J. Fourier Anal. Appl. 26 (2020), Paper No. 26).

### 2022/03/08

17:00-18:00   Room #online (Graduate School of Math. Sci. Bldg.)
Masatoshi Kitagawa (Waseda University)
On the structure of Hamiltonian G-varieties (Japanese)
[ Abstract ]
I will talk about a result by I. Losev (Math. Z. 2009) on the structure of Hamiltonian G-varieties.
In particular, I will explain how to reduce the result to central-nilpotent cases.
I will give an application of the result to branching laws.

### 2022/02/22

17:00-18:00   Room #online (Graduate School of Math. Sci. Bldg.)
Hiroyoshi Tamori (Hokkaido University)
On a long exact sequence of the Schwartz homology (Japanese)
[ Abstract ]
For a smooth Fr\’{e}chet representation of moderate growth of an almost linear Nash group, Chen-Sun introduced a homology (called Schwartz homology) equipped with certain topology. Given a short exact sequence of such representations, we can construct a long exact sequence of Schwartz homology groups via the natural isomorphism with relative Lie algebra homology. We give an example of a long exact sequence where the connecting homomorphism is not continuous.

### 2022/02/15

17:00-18:00   Room #online (Graduate School of Math. Sci. Bldg.)
Kazuki Kannaka (RIKEN iTHEMS)
Deformations of standard compact Clifford-Klein forms (Japanese)
[ Abstract ]
Let Γ be a discontinuous group for a homogeneous manifold G/H of reductive type.
The Clifford-Klein form Γ\G/H is standard if Γ is contained in a reductive subgroup of G acting properly on G/H.

For 12 series of standard compact Clifford-Klein forms given by Kobayashi-Yoshino, we discuss in this talk whether or not there exist (1) locally rigid ones, (2) non-standard deformations, and (3) Zariski-dense deformations in G.

After briefly explaining Kobayashi's work and Kassel's work on these
questions, we will explain the new results.

### 2022/01/18

17:00-18:00   Room #On line (Graduate School of Math. Sci. Bldg.)
Hideyuki Ishi (Osaka City University)
Strongly visible actions on complex domains (Japanese)
[ Abstract ]
In this century, the Cartan-Hartogs domain and its variations, on which the Bergman kernel function and the Kahler-Einstein metric can be computed explicitly, have been actively studied. Reasoning that strongly visible actions on the domains enable such nice calculations, we introduce a new type of complex domain analogous to the Cartan-Hartogs domain, and present a research plan about harmonic analysis over the domain.

### 2022/01/11

17:00-18:00   Room #on line (Graduate School of Math. Sci. Bldg.)
Joint with Tuesday Seminar on Topology.
Keiichi Maeta (The University of Tokyo)
On the existence problem of Compact Clifford-Klein forms of indecomposable pseudo-Riemannian symmetric spaces with signature (n,2) (Japanese)
[ Abstract ]
For a homogeneous space $G/H$ and its discontinuous group $\Gamma\subset G$, the double coset space $\Gamma\backslash G/H$ is called a Clifford-Klein form of $G/H$. In the study of Clifford-Klein forms, the classification of homogeneous spaces which admit compact Clifford-Klein forms is one of the important open problems, which was introduced by Toshiyuki Kobayashi in 1980s.

We consider this problem for indecomposable and reducible pseudo-Riemannian symmetric spaces with signature (n,2). We show the non-existence of compact Clifford-Klein forms for some series of symmetric spaces, and construct new compact Clifford-Klein forms of countably infinite five-dimensional pseudo-Riemannian symmetric spaces with signature (3,2).

### 2021/12/21

17:30-18:30   Room #on line (Graduate School of Math. Sci. Bldg.)
Joint with Tuesday Seminar on Topology.
Hiroki Shimakura (Tohoku University)
Classification of holomorphic vertex operator algebras of central charge 24
(Japanese)
[ Abstract ]
Holomorphic vertex operator algebras are important in vertex operator algebra theory. For example, the famous moonshine vertex operator algebra is holomorphic.
One of the fundamental problems is to classify holomorphic vertex operator algebras. It is known that holomorphic vertex operator algebras of central charge 8 and 16 are lattice vertex operator algebras.
I will talk about recent progress on the classification of holomorphic vertex operator algebras of central charge 24.

### 2021/12/14

17:00-18:00   Room #on line (Graduate School of Math. Sci. Bldg.)
Yosuke Morita (Kyoto University)
On the definition of Conley indices (Japanese)
[ Abstract ]
Conley indices are used to describe local behaviour of topological dynamical systems. In this talk, I will explain a new framework for Conley index theory. Our approach is very elementary, and uses only general topology and some computations of inclusion relations of subsets.

### 2021/12/07

17:00-18:00   Room #on line (Graduate School of Math. Sci. Bldg.)
Toshihisa Kubo (Ryukoku University)
On the classification of the $K$-type formulas for the Heisenberg ultrahyperbolic equation (Japanese)
[ Abstract ]
About ten years ago, Kable constructed a one-parameter family $\square^{(n)}_s$ ($s\in \mathbb{C}$) of differential operators for $\mathfrak{sl}(n,\mathbb{C})$. He referred to $\square^{(n)}_s$ as the Heisenberg ultrahyperbolic operator. In the viewpoint of intertwining operators, $\square^{(n)}_s$ can be thought of as an intertwining differential operator between certain parabolically induced representations for $\widetilde{SL}(n,\mathbb{R})$. In this talk we discuss about the classification of the $K$-type formulas of the space of $K$-finite solutions to the differential equation $\square^{(3)}_sf=0$ for $\widetilde{SL}(3,\mathbb{R})$ and some related topics. This is joint work with Bent {\O}rsted.

### 2021/11/30

17:00-18:00   Room #on line (Graduate School of Math. Sci. Bldg.)
Quentin Labriet (Reims University)
Branching problems for conformal Lie groups and orthogonal polynomials (English)
[ Abstract ]
In this talk, I will present some results obtained during my PhD about a link between branching problems for conformal Lie groups and orthogonal polynomials. More precisely, I am going to look at some examples of branching problems for representations in the scalar-valued holomorphic discrete series of some conformal Lie groups. Using the geometry of symmetric cone, I will explain how the theory of orthogonal polynomials can be related to branching problems and to the construction of symmetry breaking and holographic operators.

### 2021/11/23

17:00-18:00   Room #on line (Graduate School of Math. Sci. Bldg.)
Yuichiro Tanaka (The University of Tokyo)
A Cartan decomposition for a reductive real spherical subgroup
(Japanese)
[ Abstract ]
A closed subgroup H of a real reductive Lie group G is real spherical if a minimal parabolic subgroup of G has an open orbit on G/H.
In this talk I would like to show a proof of a Cartan decomposition G=KAH when H is reductive.
This is a conjecture of T. Kobayashi, introduced in the 3rd Number Theory Summer School in 1995.

### 2021/11/16

17:00-18:00   Room #on line (Graduate School of Math. Sci. Bldg.)
Ryosuke Nakahama (Kyushu University)
Computation of weighted Bergman norms on block diagonal matrices in bounded symmetric domains (Japanese)
[ Abstract ]
Let $G/K\simeq D\subset\mathfrak{p}^+$ be a Hermitian symmetric space realized as a bounded symmetric domain, and we consider the weighted Bergman space $\mathcal{H}_\lambda(D)$ on $D$.
Then the norm on each $K$-type in $\mathcal{H}_\lambda(D)$ is explicitly computed by Faraut--Kor\'anyi (1990).
In this talk, we consider the cases $\mathfrak{p}^+=\operatorname{Sym}(r,\mathbb{C})$, $M(r,\mathbb{C})$, $\operatorname{Alt}(2r,\mathbb{C})$, fix $r=r'+r''$, and decompose $\mathfrak{p}^+$ into $2\times 2$ block matrices.
Then the speaker presents the results on explicit computation of the norm of $\mathcal{H}_\lambda(D)$ on each $K'$-type in the space of polynomials on the block diagonal matrices $\mathfrak{p}^+_{11}\oplus\mathfrak{p}^+_{22}$.
Also, as an application, the speaker presents the results on Plancherel-type formulas on the branching laws for symmetric pairs $(Sp(r,\mathbb{R}),U(r',r''))$, $(U(r,r),U(r',r'')\times U(r'',r'))$, $(SO^*(4r),U(2r',2r''))$.

### 2021/11/09

17:00-18:00   Room #on line (Graduate School of Math. Sci. Bldg.)
Yoshiki Oshima (Osaka University)
On the support of Plancherel measures and the image of moment map
(Japanese)
[ Abstract ]
We see a relationship between the support of Plancherel measures for homogeneous spaces and the image of moment maps from cotangent bundles based on a joint work with Benjamin Harris.
Moreover, we discuss related problems and conjectures about inductions and restrictions for representations of Lie groups in general settings.

### 2021/11/02

17:00-18:00   Room #online (Graduate School of Math. Sci. Bldg.)
Taito TAUCHI (Kyushu University)
Difference between the distribution and hyperfunction solution spaces of an irregular holonomic D-module (Japanese)
[ Abstract ]
Let M be a holonomic D-module. Then the distribution and hyperfunction solution spaces of M coincide if M is regular. However, there is a difference between them in general if M is irregular. In this talk, we explain this phenomena taking a Whittaker functional of the principal series representation of SL(2, R) as a concrete example.

### 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)
[ 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.

### 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)
[ 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.

### 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)
[ 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.

### 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)
[ 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.

### 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)
[ 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).