Rankin-Cohen brackets on tube-type domains

正交李代数的Rankin-Cohen括号与双线性共形不变微分算子

Rankin-Cohen brackets for orthogonal Lie algebras and bilinear conformally invariant differential operators

本文利用代数傅里叶变换的李理论方法，分类了正交李代数情形下标量广义Verma模对角分支规则中的奇异向量，并应用于分类共形球面上的双线性共形不变微分算子。

Inversion of Rankin-Cohen operators via Holographic Transform

The analysis of branching problems for restriction of representations brings the concept of symmetry breaking transform and holographic transform. Symmetry breaking operators decrease the number of variables in geometric models, whereas holographic operators increase it. Various expansions in classical analysis can be interpreted as particular occurrences of these transforms. From this perspective, we investigate two remarkable families of differential operators: the Rankin-Cohen operators and the holomorphic Juhl conformally covariant operators. Then we establish for the corresponding symmetr…

超弦上的Gordan-Rankin-Cohen算子

Gordan-Rankin-Cohen operators on superstrings

本文区分了模形式空间与加权密度空间间的双微分算子分类，分类了在超弦及特定李超代数下不变的Gordan-Rankin-Cohen算子类似物。

Gordan-Rankin-Cohen算子在超维度$1\vert 1$上的加权密度空间

Gordan-Rankin-Cohen operators on the spaces of weighted densities in superdimension $1\vert 1$

研究超弦理论中超维度$1\vert 1$的加权密度空间，通过Gordan-Rankin-Cohen算子分类微分算子，解决超弦问题。

Differential symmetry breaking operators. I-Genreal theory and F-method. II-Rankin-Cohen Operators for Symmetric Pairs

Part I. We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method (F-method) based on the algebraic Fourier transform for generalized Verma modules, which characterizes differential symmetry breaking operators by means of certain systems of partial differential equations. In contrast to the setting of real flag varieties, continuous symmetry b…

Conformally Covariant Bi-Differential Operators on a Simple Real Jordan Algebra

For a simple real Jordan algebra $V,$ a family of bi-differential operators from $\mathcal{C}^\infty(V\times V)$ to $\mathcal{C}^\infty(V)$ is constructed. These operators are covariant under the rational action of the conformal group of $V.$ They generalize the classical {\em Rankin-Cohen} brackets (case $V=\mathbb{R}$).

Covariant bi-differential operators on matrix space

A family of bi-differential operators from $C^\infty\big(\Mat(m,\mathbb R)\times\Mat(m,\mathbb R)\big)$ into $C^\infty\big(\Mat(m,\mathbb R)\big)$ which are covariant for the projective action of the group $SL(2m,\mathbb R)$ on $\Mat(m,\mathbb R)$ is constructed, generalizing both the \emph{transvectants} and the \emph{Rankin-Cohen brackets} (case $m=1$).

Holographic transform for tensor product of holomorphic discrete series

We study holographic operators associated with Rankin-Cohen brackets which are symmetry breaking operators for the restriction of tensor products of holomorphic discrete series of SL2(R). Furthermore, we investigate a geometrical interpretation of these operators and their relations to classical Jacobi polynomials.

对称破缺的生成算子——从离散到连续

Generating operators of symmetry breaking -- from discrete to continuous

本研究基于既有Rankin–Cohen括号的生成算子，提出从可数微分对称破缺算子构造多种带连续参数的基本算子的方法，涵盖不变三线性形式、反德西特空间上的变换及融合规则相关算子等。

两个Verma模张量积的分支问题及其在微分对称破缺算子中的应用

Branching problem of tensoring two Verma modules and its application to differential symmetry breaking operators

本文研究半单李代数中两个Verma模张量积的分支问题，为Kobayashi-Pevzner发现的sl2情形重数一失效参数提供新刻画，并描述不可分解分量及同构条件。

Conformally covariant bi-differential operators for differential forms

The classical Rankin-Cohen brackets are bi-differential operators from $C^\infty(\mathbb R)\times C^\infty(\mathbb R)$ into $ C^\infty(\mathbb R)$. They are covariant for the (diagonal) action of ${\rm SL}(2,\mathbb R)$ through principal series representations. We construct generalizations of these operators, replacing $\mathbb R$ by $\mathbb R^n,$ the group ${\rm SL}(2,\mathbb R)$ by the group ${\rm SO}_0(1,n+1)$ viewed as the conformal group of $\mathbb R^n,$ and functions by differential forms.

小$q$-雅可比多项式与$U_q(sl_2)$的对称破毁算子

Little $q$-Jacobi polynomials and symmetry breaking operators for $U_q(sl_2)$

本文研究了$U_q(sl_2)$量子群作用于张量积Verma模的 intertwining 算子，利用小$q$-雅可比多项式和$q$-变形的Rankin-Cohen算子，揭示了非交换变量与Clebsch-Gordan系数之间的联系。