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
AI总结：
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。
摘要：
基于代数傅里叶变换的李理论方法，我们在诱导参数取一般值的情况下，对正交李代数及其具有交换幂零根的共形抛物子代数情形下，标量广义Verma模的对角分支规则所对应的标量奇异向量进行了分类，从而以显式方式实现了对角分支规则。奇异向量的复杂组合结构可通过广义超几何函数${}_3F_2$的递推关系方便地确定。作为几何应用，我们分类了作用在由共形球面$S^n$给出的旗流形上的齐次线丛上的双线性共形等变微分算子。
英文摘要：
Based on the Lie theoretical methods of algebraic Fourier transformation, we classify in the case of generic values of inducing parameters the scalar singular vectors corresponding to the diagonal branching rules for scalar generalized Verma modules in the case of orthogonal Lie algebra and its conformal parabolic subalgebra with commutative nilradical, thereby realizing the diagonal branching rules in an explicit way. The complicated combinatorial structure of singular vectors is conveniently determined in terms of recursion relations for the generalized hypergeometric function ${}_3F_2$. As a geometrical application, we classify bilinear conformally equivariant differential operators acting on homogeneous line bundles on the flag manifold given by conformal sphere $S^n$.
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