On the cohomology of line bundles over certain flag schemes

Abstract

Let $G$ be the group scheme ${\mathrm{SL}}_{d+1}$ over $\mathbb{Z}$ and let $Q$ be the parabolic subgroup scheme corresponding to the simple roots ${\alpha }_2,\cdots ,{\alpha }_{d-1}$. Then $G/Q$ is the $\mathbb{Z}$-scheme of partial flags $\{D_1\subset H_d\subset V\}$. We will calculate the cohomology modules of line bundles over this flag scheme. We will prove that the only non-trivial ones are isomorphic to the kernel or the cokernel of certain matrices with multinomial coefficients.