Cohomology of Line Bundles on the Flag Variety for Type G_2
  In the case of an almost simple algebraic group $G$ of type $G_2$ 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 $p^2$-alcoves.   When $U_q$ 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. 
Henning Haahr Andersen, Masaharu Kaneda
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