Cohomology of line bundles on the flag variety for type G2

In the case of a simple algebraic group G of type G2 over a field of characteristic p>0 we study the cohomology modules of line bundles on the flag variety for G. Our main result is a complete determination of the vanishing behavior of such cohomology in the case where the line bundles in question are induced by characters from the lowest p2-alcoves.When Uq is the quantum group corresponding to G whose parameter q is a complex root of unity of order prime to 6 we give a complete (i.e. covering all characters) description of the vanishing behavior for the corresponding quantized cohomology modules.