Positive characteristic[edit]

One also has a weaker form of this theorem in positive characteristic. Namely, let G be a semisimple algebraic group over an algebraically closed field of characteristic ${\displaystyle p>0}$. Then it remains true that ${\displaystyle H^{i}(G/B,\,L_{\lambda })=0}$ for all i if λ is a weight such that ${\displaystyle w*\lambda }$ is non-dominant for all ${\displaystyle w\in W}$ as long as λ is "close to zero".^[1] This is known as the Kempf vanishing theorem. However, the other statements of the theorem do not remain valid in this setting.


More explicitly, let λ be a dominant integral weight; then it is still true that ${\displaystyle H^{i}(G/B,\,L_{\lambda })=0}$ for all ${\displaystyle i>0}$, but it is no longer true that this G-module is simple in general, although it does contain the unique highest weight module of highest weight λ as a G-submodule. If λ is an arbitrary integral weight, it is in fact a large unsolved problem in representation theory to describe the cohomology modules ${\displaystyle H^{i}(G/B,\,L_{\lambda })}$ in general. Unlike over ${\displaystyle \mathbb {C} }$, Mumford gave an example showing that it need not be the case for a fixed λ that these modules are all zero except in a single degree i.