Differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces

In this paper we classify and construct differential symmetry breaking operators $\mathbb{D}$ from a line bundle over the real projective space $\mathbb{R}{\mathbb{P}}^n$ to a vector bundle over $\mathbb{R}{\mathbb{P}}^{n-1}$. We further determine the factorization identities of $\mathbb{D}$ and the branching laws of the corresponding generalized Verma modules of $\mathfrak{s}\mathfrak{l}(n+1,\mathbb{C})$. By utilizing the factorization identities, the $SL(n,\mathbb{R})$-representations realized on the image $\text{Im}(\mathbb{D})$ are also investigated.