Wentao Teng, Takayuki Okuda, Mamoru Ueda, Masatoshi Kitagawa,
| 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。 |
© Toshiyuki Kobayashi