root . Also, we obtain the classical Borel–Weil theorem as a special case of this theorem by taking to be dominant and to be the identity element . Example For example, consider , for which is the Riemann sphere, an integral weight is specified simply by an integer , and . The line bundle is , whose sections are the homogeneous polynomials of degree (i.e. the binary forms). As a representation of , the sections can be written as , and is canonically isomorphic to . This gives us at a stroke the representation theory of : is the standard representation, and is its th symmetric power. We even have a unified description of the action of the Lie algebra, derived from its realization as vector fields on the Riemann sphere: if , , are the standard generators of , then Positive characteristic One also has a weaker form of this theorem in positive characteristic. Namely, let be a semisimple algebraic group over an algebraically closed field of characteristic . Then it remains true that for all if is a weight such that is non-dominant for all as long as is "close to zero". 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 be a dominant integral weight; then it is still true that for all , but it is no longer true that this -module is simple in general, although it does contain the unique highest weight module of highest weight as a -submodule. If is an arbitrary integral weight, it is in fact a large unsolved problem in representation theory to describe the cohomology modules in general. Unlike over , Mumford gave an example showing that it need not be the case for a fixed that these modules are all zero except in a single degree . Borel–Weil theorem The Borel–Weil theorem provides a concrete model for irreducible representations of compact Lie groups and irreducible holomorphic representations of complex semisimple Lie groups. These representations are realized in the spaces of global sections of holomorphic line bundles on the flag manifold of the group. The Borel–Weil–Bott theorem is its generalization to higher cohomology spaces. The theorem dates back to the early 1950s and can be found in and . Statement of the theorem The theorem can be stated either for a complex semisimple