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

Abstract

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