The ordered F-system for the F-method and differential symmetry breaking operators for $(GL(3,\mathbb{R}), GL(2,\mathbb{R}))$

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.