On the intertwining differential operators from a line bundle to a vector bundle over the real projective space

Abstract

We classify and construct $SL(n,\mathbb{R})$-intertwining differential operators $\mathcal{D}$ from a line bundle to a vector bundle over the real projective space $\mathbb{RP}^{n-1}$ by the F-method. This generalizes a classical result of Bol for $SL(2,\mathbb{R})$. Further, we classify the $K$-type formulas for the kernel $\text{Ker}(\mathcal{D})$ and image $\text{Im}(\mathcal{D})$ of $\mathcal{D}$. The standardness of the homomorphisms $\varphi$ corresponding to the differential operators $\mathcal{D}$ between generalized Verma modules are also discussed.