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Lie Groups and Representation Theory Seminar 2021

List of speakers:
Ryosuke Nakahama, Mamoru Ueda, Masatoshi Kitagawa, Kazuki Kannaka, Toshiyuki Kobayashi, Yosuke Morita, Hidenori Fujiwara, Taito Tauchi, Yoshiki Oshima #1, Hiroyoshi Tamori, Yoshiki Oshima #2,
Date: May 11 (Tue), 2021, 17:00-18:00
Speaker: Ryosuke Nakahama (中濱良祐) (Kyushu University)
Title: 有界対称領域上の重み付きベルグマン内積の計算と部分群への制限 / Computation of weighted Bergman inner products on bounded symmetric domains and restriction to subgroups
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$D \subset M(r,\mathbb{C})$を有界対称領域とし, その上の重み付きベルグマン空間$\mathcal{H}_{\lambda} (D)$を考える. すると,ここに$SU(r,r)$がユニタリに作用する.本セミナーでは, $\operatorname{Alt}(r,\mathbb{C})$, $\operatorname{Sym}(r,\mathbb{C}) \subset M(r,\mathbb{C})$上のある多項式について,内積を具体的に計算し, また特にこの多項式が行列式またはパフ式の冪の場合には 内積が多変数超幾何多項式で与えられることを示す. またこの応用として,$SU(r,r)$から$Sp(r,\mathbb{R})$または $SO^*(2r)$への対称性破れ作用素の構成に関する結果を述べる.

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 $SO^*(2r)$.

Date: May 18 (Tue), 2021, 17:00-18:30
Speaker: Mamoru Ueda (上田衛) (Kyoto University)
Title: アファインヤンギアンと長方形W代数 / Affine Yangians and rectangular W-algebras
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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.
Joint with Tuesday Seminar on Topology
Date: June 1 (Tue), 2021, 17:30-18:30
Speaker: Masatoshi Kitagawa (北川 宜稔) (Waseda University)
Title: On the discrete decomposability and invariants of representations of real reductive Lie groups
[ pdf ]
群の既約表現を部分群に制限したときにどのように振る舞うかを記述する問題を 分岐則の問題という。 既約表現の制限は一般には既約ではなくなり、ユニタリな場合には直積分で記述 される既約分解が存在する。 この分解は、ユニタリ作用素のスペクトル分解の一般化とみなすことができ、一 般には連続的なスペクトルと離散的なスペクトルが現れる。 連続的なスペクトルが現れない場合、つまりユニタリ表現の離散的な直和になっ ている場合、その表現は離散分解するという。

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.

Date: June 8 (Tue), 2021, 17:00-18:00
Speaker: Kazuki Kannaka (甘中一輝) (RIKEN)
Title: 3次元コンパクト反ド・ジッター多様体の安定固有値の重複度 / The multiplicities of stable eigenvalues on compact anti-de Sitter 3-manifolds
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擬リーマン局所対称空間とは 半単純対称空間$G/H$の不連続群$\Gamma$による商多様体$\Gamma\backslash G/H$の事である。 小林俊行は擬リーマン局所対称空間の(ラプラシアンのような)内在的微分作用素のスペクトル解析 の研究を創始した。古典的なリーマン多様体の設定とは異なり、 擬リーマン局所対称空間のラプラシアンはもはや楕円型微分作用素ではない。 そのスペクトル解析において、Kassel・小林による先駆的研究に続き、 リーマン多様体の設定とは異なる新たな現象が近年発見されつつある。 例えば、Kassel・小林は$\Gamma$を変形させた時の $\Gamma\backslash G/H$の内在的微分作用素の固有値の振る舞い を研究した。特別な場合として$3$次元コンパクト反ド・ジッター多様体 $\Gamma\backslash \mathrm{SO}(2,2)/\mathrm{SO}(2,1)$ の(双曲型)ラプラシアンの無限個の安定固有値を発見した([Adv. Math. 2016])。 本講演では、反ド・ジッター多様体の設定で 安定固有値の重複度についての最近の結果について説明したい。

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.

Date: June 15 (Tue), 2021, 17:00-18:00
Speaker: Toshiyuki Kobayashi (小林俊行) (The Univ. of Tokyo)
Title: 極限代数と緩増加表現 / Limit algebras and tempered representations
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次の4つの(見かけ上は無関係な)4つのトピックについての新しい関係 について話す予定です。
  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
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".
Date: June 22 (Tue), 2021, 17:00-18:00
Speaker: Yosuke Morita (森田陽介) (Kyoto University)
Title: 非簡約な部分群のCartan射影とコンパクトClifford-Klein形の存在問題 / Cartan projections of some nonreductive subgroups and the existence problem of compact Clifford-Klein forms
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$G$を簡約Lie群、$H$を$G$の閉部分群、$\Gamma$を$G$の離散部分群とする。 Benoist-小林の判定法によれば、$\Gamma$の$G/H$への作用の固有性は、 $H$と$\Gamma$のCartan射影によって決定される。 非簡約な部分群のCartan射影は大抵計算が困難だが、 中には具体的に計算可能な例もある。 そうした部分群を利用して、コンパクトなClifford-Klein形を持たない簡約型等質空間の例が得られることを紹介する。
Let $G$ be a reductive Lie group, $H$ a closed subgroup of $G$, and $\Gamma$ a discrete subgroup of $G$. According to the Benoist-Kobayashi criterion, the properness of the $\Gamma$-action on $G/H$ is determined by the Cartan projections of H and $\Gamma$. 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.
Date: June 29 (Tue), 2021, 17:00-18:00
Speaker: Hidenori Fujiwara (藤原英徳)
Title: 冪零リー群に対する多項式予想について / Polynomial conjectures for nilpotent Lie groups
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G = exp g をリー環 g をもつ連結・単連結な冪零リー群とし、H = exp h をリー環 h をもつ G の解析部分群、χ を H のユニタリ指標とし、G の単項表現 τ = ind_H^G χ を考える。このとき、τ の既約分解における重複度は一様に有界であるかまたは一様に ∞ に等しいことが知られている。前者の場合 τ は有限重複度をもつという。 さて、データ (H,χ) に伴う G/H 上の直線束に作用する G-不変微分作用素の環を D_τ(G/H) で表す。τ が有限重複度をもつことと D_τ(G/H) が可換であることは同値である。1992 年 Corwin-Greenleaf は次の多項式予想を提出した: τ が有限重複度をもつとき、環 D_τ(G/H) は Γ_τ 上の H-不変多項式環C[Γ_τ]^H と同型であろう。ここでΓ_τ は g の双対ベクトル空間の元で h への制限が-√-1 dχ を満たすものがなすアファイン部分空間である。 群の表現論において2つの操作,誘導と制限,の間にはある種の双対性があることが良く知られている。そこで表現の制限についても多項式予想を考えてみよう。G をこれまで通り連結・単連結な冪零リー群、π をその既約ユニタリ表現とする。K を G の解析部分群とし π の K への制限 π|_K を考える。今回もまた π|_K の既約分解における重複度は一様に有界であるかまたは一様に ∞に等しいことが知られている。前者の場合 π|_K は有限重複度をもつといい、我々はこれを仮定しよう。G のリー環 g の複素化の包絡環を U(g) とし、不変微分作用素環 (U((g)/ker π)^Kを考える。つまり K-不変な元の全体である。すると、π| Kが有限重複度をもつことと (U((g)/ker π)^K が可換環であることは同値である。このとき環 (U((g)/ker π)^K は Ω(π) 上の K-不変多項式環C[Ω(π)]^K と同型であろうか? ここで Ω(π) は π に対応する G の余随伴軌道である。 我々はこれら 2 つの多項式予想を証明したい。
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.
Date: July 6 (Tue), 2021, 17:00-18:00
Speaker: Taito Tauchi (田内大渡) (Kyushu University)
Title: Casselmanの部分表現定理に関するQシリーズ類似の反例について / A counterexample to a Q-series analogue of Casselman's subrepresentation theorem
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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 $Q$.

Date: July 13 (Tue), 2021, 17:00-18:00
Speaker: Yoshiki Ohima (大島芳樹) (Osaka University)
Title: 局所対称空間のコンパクト化と自然なKahler計量の崩壊 / Compactification of locally symmetric spaces and collapsing of canonical Kahler metrics
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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.

Date: July 20 (Tue), 2021, 17:00-18:00
Speaker: Hiroyoshi Tamori (田森宥好) (Hokkaido University)
Title: 零でない線形周期の存在の必要条件 / On the existence of a nonzero linear period
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$(G,H)$を対称対$(\mathrm{GL}(n,\mathbb{H}),\mathrm{GL}(n,\mathbb{C})), (\mathrm{GL}(2n,\mathbb{R}),\mathrm{GL}(n,\mathbb{C}))$とする. この時,$G$の滑らかで緩増加な既約認容Fréchet表現$\pi$の $H$-線形周期の空間の次元は$1$以下であることがBroussous-Matringeにより知られている. $G$の旗多様体の各$H$-軌道が主系列表現のホモロジーに与える影響を考えることで, $\pi$の零でない$H$-線形周期が存在する必要条件を紹介する. これはアルキメデス局所体の場合のPrasadとTakloo-Bighashの予想を与える. 鈴木美裕氏(金沢大学)との共同研究に基づく.

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).

Date: July 28 (Wed), 2021, 17:00-18:00
Speaker: Yoshiki Ohima (大島芳樹) (Osaka University)
Title: Ricci平坦計量の崩壊とMonge-Ampere方程式の解のアプリオリ評価 / Collapsing Ricci-flat metrics and a priori estimate for the Monge-Ampere equation
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YauはMonge-Ampere方程式の解のアプリオリ評価を行ってCalabi予想を証明した. 近年ファイバー空間の構造を持つCalabi-Yau多様体 について,底空間のKahler類に崩壊するようなRicci平坦Kahler計量の振舞が Gross-Tosatti-Zhang等により研究されている.尾高悠志 との共同研究(arXiv:1810.07685)で得られたK3曲面の球面へのGromov-Hausdorff収束も,これらのMonge-Ampere方程式の解の評価に基 づいている.この講演では,微分方程式の解の評価がどのように自然な計量の存 在やGromov-Hausdorff収束を導くかをお話ししたい.

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.

© Toshiyuki Kobayashi