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In the case of an almost simple algebraic group $G$ of type $G_2$ over afield of characteristic $p>0$ we study the cohomology modules of line bundleson the flag variety for $G$. Our main result is a complete determination of thevanishing behavior of such cohomology in the case where the line bundles inquestion are induced by characters from the lowest $p^2$-alcoves.  When $U_q$ is the quantum group corresponding to $G$ whose parameter $q$ is acomplex root of unity of order prime to 6 we give a complete (i.e. covering allcharacters) description of the vanishing behavior for the correspondingquantized cohomology modules.

Cohomology of Line Bundles on the Flag Variety for Type G_2