Cohomology of line bundles on flag varieties over complex numbers is a classical topic in the representation theory of complex semisimple algebraic groups. On the contrary, in positive characteristic these cohomology groups are much harder to compute, and little is known beside the case of groups of rank two. In this talk, we are going to explain how the knowledge of cohomology of line bundles on flag varieties in positive characteristic can be used to compute the cohomology of the sheaf of differential operators with divided powers. In particular, we will show how the higher cohomology vanishing leads to localization theorems for D-modules in positive characteristic. We will illustrate the method on the example of groups of rank two, the case of the exceptional group G_2 being the most interesting.