Cohomology of Line Bundles on the Flag Variety for Type G_2

TL;DR

This paper investigates the cohomology of line bundles on the flag variety for the algebraic group of type G_2 over fields of positive characteristic and quantum groups at roots of unity, providing complete vanishing criteria.

Contribution

It provides a full characterization of the vanishing behavior of cohomology modules for line bundles induced by characters from the lowest p^2-alcoves in characteristic p>0 and for quantum groups at roots of unity.

Findings

Complete determination of cohomology vanishing in characteristic p>0.

Full description of quantum cohomology vanishing at roots of unity.

Results applicable to all characters in the specified alcoves.

Abstract

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.