AI 中文总结
本文针对非阿贝尔幂根基的F-方法引入有序F-系统，统一处理微分对称破缺算子的对称化与有序形式，完成$(GL(3,\boldsymbol{\reals}), GL(2,\boldsymbol{\reals}))$主系列表示间DSBO的分类构造，相关结果由Verma模分支律支撑。
AI 中文摘要
本文介绍了非幂根基未必为阿贝尔时产生的F-方法对应的F-系统的新侧面。为此，我们在有限维李代数$\boldsymbol{\frakg}$的对偶空间$\boldsymbol{\frakg}^\boldsymbol{\text{v}}$上的多项式函数空间上定义了对称化算子。在F-方法的语境下，用该算子对F-系统做共轭变换可得到新系统，我们称之为有序F-系统。这两种技术使人们能够统一处理微分对称破缺算子（DSBO）的对称化形式（带对称化）和有序形式（不带对称化）。作为该理论的应用，我们对$(GL(3,\boldsymbol{\reals}), GL(2,\boldsymbol{\reals}))$的主系列表示之间的所有DSBO进行了分类与构造，涵盖对称化和有序两种形式。此处，我们考虑对应于$\boldsymbol{\frakg l}(3,\boldsymbol{\reals})$正根的所有嵌入$GL(2,\boldsymbol{\reals}) \to GL(3,\boldsymbol{\reals})$。此外，我们利用上述李群对的DSBO研究$GL(3,\boldsymbol{\reals})$的微分交错算子（DIO）和$(SL(3,\boldsymbol{\reals}), SL(2,\boldsymbol{\reals}))$的DSBO，还研究了对应Verma模的分支律以支撑DSBO的结果。
英文摘要
In this paper, we introduce a new aspect of the F-system for the F-method arising in the case where the nilpotent radical is not necessarily abelian. For this, we define the symmetrization operator on the space of polynomial functions on the dual $\mathfrak{g}^\vee$ of a finite-dimensional Lie algebra $\mathfrak{g}$. In the context of the F-method, the conjugation by this operator of the F-system yields a new system, which we call the ordered F-system. These two techniques allow one to consider the symmetrized form (with symmetrization) and ordered form (without symmetrization) of differential symmetry breaking operators (DSBOs) uniformly. As an application of this theory, we classify and construct all the DSBOs between principal series representations for $(GL(3,\mathbb{R}), GL(2,\mathbb{R}))$ in both symmetrized and ordered forms. Here, we consider all embeddings $GL(2,\mathbb{R}) \hookrightarrow GL(3,\mathbb{R})$ corresponding to the positive roots of $\mathfrak{gl}(3,\mathbb{R})$. Furthermore, we utilize the DSBOs for the above pair to investigate differential intertwining operators (DIOs) for $GL(3,\mathbb{R})$ and DSBOs for $(SL(3,\mathbb{R}), SL(2,\mathbb{R}))$. The branching laws of the corresponding Verma modules are also studied to support our results of DSBOs.
