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 $SO^*(2r)$.

離散分解する分岐則は技術的に扱いやすいというだけでなく、大きな群の表現の 情報から小さい部分群の表現の情報が取り出しやすい状況になっており、 以下のような応用が知られている。 保型形式から別の保型形式を作り出す Rankin--Cohen ブラケットという作用素は、 離散分解する表現から既約表現への絡作用素として得られることが知られており、 近年でも多くの一般化が得られている。 また、等質空間の $L^2$ 関数の空間の離散スペクトルを別の等質空間のものから構 成するという結果にも用いられている。(T. Kobayashi, J. Funct. Anal. ('98))

本講演では、実簡約リー群の既約表現の制限の離散分解性について、小林俊行氏 が提唱した離散分解性と$G'$-許容性の一般論と判定条件 (Invent. math. '94, Annals of Math. '98, Invent. math. '98)を踏まえつつ、 最近得られた結果を紹介したい。 表現の代数的な不変量である随伴多様体、解析的な不変量である wave front set、表現空間の位相、の三つを用いて離散分解性を記述する。

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 \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.

1. （解析）　等質空間上のユニタリ表現が緩増加
2. （組合せ論）　凸多面体
3. （トポロジー）極限代数　
4. (シンプレクティック幾何学）　余随伴軌道の幾何学的量子化

1. (analysis) Tempered unitary representations on homogeneous spaces
2. (combinatorics) Convex polyhedral cones
3. (topology) Limit algebras
4. (symplectic geometry) Quantization of coadjoint orbits

Let $G$ be a real reductive Lie group, $Q$ a parabolic subgroup of $G$, and $\pi$ an irreducible admissible representation of $G$. We say that $\pi$ 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 $Q$.

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.

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 Fréchet 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).

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.

Applying the criterion to symmetric pairs, we give a full description of the triples $H \subset G \supset G'$ 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.

実簡約リー群の "小さな" 無限次元表現の族に対して、部分群への分岐則の重複度がいつ有界になるか、に関する幾何的な判定条件を説明する。

この幾何的な必要十分条件を$G/H$と$G/G$が共に簡約対称空間の場合に適用し、 $H$-distinguishedな$G$の任意の既約表現が、部分群$G'$の表現として有界重複性を もつための3つ組 $H \subset G \supset G'$を完全に分類することができる（[Adv. Math. 2021, 7節]、arXiv:2109.14424）

これらの結果について、できるだけわかりやすく解説する予定である。

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.

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.

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 phenomenon taking a Whittaker functional of the principal series representation of $\mathit{SL}(2, \mathbb{R})$ as a concrete example.

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.

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-Korányi (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 $(Sp(r,\mathbb{R}),U(r',r''))$, $(U(r,r),U(r',r'')\times U(r'',r'))$, $(SO^*(4r),U(2r',2r''))$.

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.

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 Ørsted.