Algebraic construction of contragradient quasi-Verma modules in positive characteristic

In the present paper we investigate a new class of infinite-dimensional modules over the hyperalgebra of a semi-simple algebraic group in positive chararacteristic called quasi-Verma modules. We provide a purely algebraic construction of the global Grothendieck-Cousin complex corresponding to the standard line bumdle ${\mathcal L}(\lambda)$ on the Flag variety of the algebraic group stratified by Schubert cells. We prove that the complex consists of direct sums of quasi-Verma modules for the highest weights of the form $w\cdot\lambda$ for various elements $w$ of the Weyl group.