Wentao Teng, Takayuki Okuda, Mamoru Ueda, Masatoshi Kitagawa, Paolo Ciatti, Valentina Casarino, Koichi Arashi, Kazuki Kannaka,
| Date: | Mar 21 (Fri), 2025, 17:00-17:40 |
| Speaker: | Wentao Teng (The University of Tokyo) |
| Title: | A positive product formula of integral kernels of $k$-Hankel transforms |
| Abstract: [ pdf ] | Let $R$ be a root system in $\mathbb R^N$ and $G$ be the finite subgroup generated by the reflections associated to the root system. We establish a positive radial product formula for the integral kernels $B_{k,1}(x,y)$ of $(k,1)$-generalized Fourier transforms (or the $k$-Hankel transforms) $F_{k,1}$ \[ B_{k,1}(x,z)j_{2\left\langle k\right\rangle+N-2}\left(2\sqrt{t\left|z\right|}\right)=\int_{\mathbb R^N} B_{k,1}(\xi,z)\,d\sigma_{x,t}^{k,1}(\xi), \] where $j_{\lambda}$ is the normalized Bessel function, and $\sigma_{x,t}^{k,1}(\xi)$ is the unique probability measure. Such a product formula is equivalent to the following representation of the generalized spherical mean operator $f\mapsto M_f,\;f\in C_b(\mathbb{R}^N)$ in $(k,1)$-generalized Fourier analysis \begin{align*} M_f(x,t)=\int_{\mathbb{R}^N}f\,d\sigma_{x,t}^{k,1},\;(x,t) \in\mathbb{R}^N\times{\mathbb{R}}_+. \end{align*} We will then analyze the representing measure $\sigma_{x,t}^{k,1}(\xi)$ and show that the support of the measure is contained in \[ \left\{\xi\in\mathbb{R}^N:\sqrt{\vert\xi\vert}\geq\vert\sqrt{\vert x\vert}-\sqrt t\vert\right\}\cap\left(\bigcup_{g\in G}\{\xi\in\mathbb{R}^N:d(\xi,gx)\leq\sqrt t\}\right), \] where $d\left(x,y\right)=\sqrt{\left|x\right|+\left|y\right|-\sqrt{2\left(\left|x\right|\left|y\right|+\left\langle x,y\right\rangle\right)}}$. Based on the support of the representing measure $\sigma_{x,t}^{k,1}$, we will get a weak Huygens's principle for the deformed wave equation in $(k,1)$-generalized Fourier analysis. Moreover, for $N\geq 2$, if we assume that $F_{k,1}\left(\mathcal S(\mathbb{R}^N)\right)$ consists of rapidly decreasing functions at infinity, then we get two different results on $\text{supp}\sigma_{x,t}^{k,1}$, which indirectly denies such assumption. |
| Joint with Tuesday Seminar on Topology | |
| Date: | Apr 22 (Tue), 2025, 17:30-18:30 |
| Speaker: | Takayuki Okuda (奥田 隆幸) (Hiroshima University) |
| Title: | Coarse coding theory and discontinuous groups on homogeneous spaces |
| Abstract: [ pdf ] | Let $M$ and $\mathcal{I}$ be sets, and consider a surjective map \[ R : M \times M \to \mathcal{I}. \] For each subset $\mathcal{A} \subseteq \mathcal{I}$, we define $\mathcal{A}$-free codes on $M$ as subsets $C \subseteq M$ satisfying \[ R(C \times C) \cap \mathcal{A} = \emptyset. \] This definition encompasses various types of codes, including error-correcting codes, spherical codes, and those defined on association schemes or homogeneous spaces. In this talk, we introduce a "pre-bornological coarse structure" on $\mathcal{I}$ and define the notion of coarsely $\mathcal{A}$-free codes on $M$. This extends the concept of $\mathcal{A}$-free codes introduced above. As a main result, we establish relationships between coarse coding theory on Riemannian homogeneous spaces $M = G/K$ and discontinuous group theory on non-Riemannian homogeneous spaces $X = G/H$. |
| Date: | May 13 (Tue), 2025, 15:45-16:45 |
| Speaker: | Mamoru Ueda (上田 衛) (The University of Tokyo) |
| Title: | Affine Yangians and non-rectangular W-algebras of type A / アファインヤンギアンと非長方形型W代数 |
| Abstract: [ pdf ] |
The Yangian is a quantum group introduced by Drinfeld and is a
deformation of the current Lie algebra in finite setting. Yangians are
actively used for studies of one kind of vertex algebra called a
W-algebra. One of the representative results is that Brundan and
Kleshchev wrote down a finite W-algebra of type A as a quotient
algebra of the shifted Yangian. The shifted Yangian contains a finite
Yangian of type A as a subalgebra. De Sole, Kac, and Valeri
constructed a homomorphism from this subalgebra to the finite
W-algebra of type A by using the Lax operator.
In this talk, I will explain how to construct a homomorphism from the affine Yangian of type A to a non-rectangular W-algebra of type A, which can be regarded as an affine version of the result of De Sole-Kac-Valeri. This homomorphism is expected to lead to a generalization of the AGT conjecture. ヤンギアンはDrinfeldにより導入された量子群であり、有限型の場合にはカレントリー代数の変形となる。近年、ヤンギアンは頂点代数の一種であるW代数の研究で重要な役割を果たしている。 その代表的な成果の一つとして、BrundanとKleshchevがA型有限W代数をシフト型ヤンギアンの商代数として書き下したことで挙げられる。シフト型ヤンギアンはA型有限型ヤンギアンを部分代数として含んでいる。De Sole-Kac-ValeriはLax作用素を用いてこの部分代数からA型有限W代数への写像を構成した。 本講演では、De Sole-Kac-Valeriの結果のアファイン版に相当する、A型アファインヤンギアンからA型非長方形型W代数への写像を構成する方法について解説する。この写像は、AGT予想の一般化に繋がると期待されている。 |
| Date: | May 20 (Tue), 2025, 15:30-16:30 |
| Speaker: | Masatoshi Kitagawa (北川宜稔) (Kyushu University) |
| Title: | On the restriction of good filtration in the branching problem of reductive Lie groups / 簡約リー群の分岐則におけるgood filtrationの制限について |
| Abstract: [ pdf ] |
In arXiv:2405.10382, a Cartan subalgebra related to the branching
problem of reductive Lie groups was defined.
It is considered to control the size and shape of the continuous
spectrum in irreducible decompositions, and is defined using the support
of the action of the center of the universal enveloping algebra.
Except in special cases, direct computations from the definition of this
Cartan subalgebra are difficult.
In this talk, I will present results on restrictions of good filtrations and show a relation between the associated varieties of representations and the Cartan subalgebra. I will also discuss applications to the necessary condition for discrete decomposability and related conjectures by T. Kobayashi. arXiv:2405.10382において、簡約リー群の分岐則と関係するカルタン部分代数を定義した。 これは既約分解の連続スペクトルの大きさや形を統制するものと考えられ、普遍包絡環の中心の作用の台を使って定義される。 特別な場合を除き、このカルタン部分代数の定義からの直接計算は困難である。 本講演では、good filtrationの制限に関する結果を述べ、表現の随伴多様体とカルタン部分代数を関連付ける結果を示す。 また、小林俊行氏による離散分解性の必要条件と関連する予想への応用についても紹介する。 |
| Date: | June 10 (Tue), 2025, 15:30-16:30 |
| Speaker: | Paolo Ciatti (University of Padua) |
| Title: | Spectral estimates on the Heisenberg group |
| Abstract: [ pdf ] | In this talk we will discuss some estimates concerning the spectral projections of the sub-Laplacian on the Heisenberg group. We will also consider some open problems and formulate a conjecture, providing some motivation for it. |
| Date: | June 11 (Wed), 2025, 14:00-15:00 |
| Speaker: | Valentina Casarino (University of Padua) |
| Title: | Variational inequalities in a nonsymmetric Gaussian framework |
| Abstract: [ pdf ] | In this talk we introduce variation seminorms and consider the variation operator of a nonsymmetric Ornstein--Uhlenbeck semigroup $(H_ t)_{(t> 0)}$, taken with respect to $t$, in $\mathbb{R}^n$. We prove that this seminorm defines an operator of weak type $(1, 1)$ for the invariant measure. The talk is based on joint work with Paolo Ciatti (University of Padua) and Peter Sjögren (Chalmers University). |
| Date: | July 8 (Tue), 2025, 16:00-17:00 |
| Speaker: | Koichi ARASHI (嵐 晃一) (Tokyo Gakugei University) |
| Title: | On integral representations of reproducing kernels on quasi-symmetric Siegel domains / 擬対称領域上の再生核の積分表示について |
| Abstract: [ pdf ] | L. Schwartz established the foundational theory of reproducing kernels in the 1960s. Around the same time, S. G. Gindikin obtained an explicit integral representation of the Bergman kernel for the Siegel domain of the second kind $\mathcal{S}(\Omega,Q)\subset U_{\mathbb C}\times V$. This formula suggests that the set of irreducible unitary representations of the generalized Heisenberg group $G^{V}=U\rtimes V$ realized on this domain is embedded in the unitary dual of the group. Such a notion of multiplicity-freeness property has since been reconsidered from a complex-geometric standpoint, motivated by Huckleberry-Wurzbacher's study of “coisotropic actions” and by T. Kobayashi's introduction of “visible actions”, and its understanding continues to deepen. In this talk, we focus on a quasi-symmetric Siegel domain, and for a real subspace $W\subset V$, study the representations of the subgroup $G^{W}=U\rtimes W$. We show that the multiplicity-freeness property can be characterized both by geometric features of the group action and by the multiplicity-free irreducible decomposition of the unitary representation on the Bergman space. |
| Date: | July 15 (Tue), 2025, 14:30-15:30 |
| Speaker: | Kazuki KANNAKA(甘中一輝) (Kanazawa University) |
| Title: | Zariski-dense deformations of standard discontinuous groups for pseudo-Riemannian homogeneous spaces / 擬リーマン等質空間のスタンダードな不連続群のザリスキ稠密な変形 |
| Abstract: [ pdf ] |
In higher-dimensional Riemannian compact locally symmetric spaces,
rigidity theory has been developed by Selberg, Weil, Mostow, Margulis,
and so on. On the other hand, since the late 1980s, Toshiyuki Kobayashi
initiated the study of deformation theory for locally symmetric spaces
beyond the Riemannian setting. In particular, a family of pseudo-
Riemannian compact locally symmetric spaces of arbitrarily high
dimension without local rigidity were discovered. In this talk, we focus
on a class of pseudo-Riemannian compact locally symmetric spaces known
as standard ones. We explore questions such as the following: (1) Do
they possess local rigidity? (2) Can they be continuously deformed into
non-standard ones? For example, we show that compact space forms of
constant negative curvature with signature (4, 3) in dimension 7 admit
continuous deformations, analogous to hyperbolic compact Riemann
surfaces. These deformations are constructed using the bending
construction, originally introduced by Thurston. This talk is based on
joint work (arXiv:2507.03476) with Toshiyuki Kobayashi. 高次元のコンパクトなリーマン局所対称空間では, Selberg, Weil, Mostow, Margulis, ...と系譜が続く剛性理論が発展している。一方で, リーマン多様体と は限らない設定での局所対称空間の変形理論が, 1980年代後半から小林俊行氏に より研究が開始された。 特に, 局所剛性を持たない任意に高い次元を持つコンパ クトな擬リーマン局所対称空間の族が発見された。 本講演では, コンパクトな擬 リーマン局所対称空間の内, スタンダードと呼ばれるクラスのものに注目する。 そして, それらが (1) 局所剛性を持つか？ (2) スタンダードではないものに連 続変形できるか？等の問題を考察する。 例えば, 7次元の符号(4, 3)の擬リーマ ン計量を持つコンパクトな負の定曲率空間形は, 双曲型閉リーマン面の様に連続 的に変形可能である事を見る。 また, その連続変形はThurstonに由来する bending construction を用いて為される。 本講演は小林俊行氏との共同研究 (arXiv:2507.03476)に基づくものである。 |
© Toshiyuki Kobayashi