In mathematics, the Borel–Weil–Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and, more generally, from higher sheaf cohomology groups associated to such bundles. It is built on the earlier Borel–Weil theorem of Armand Borel and André Weil, dealing just with the space of sections (the zeroth cohomology group), the extension to higher cohomology groups being provided by Raoul Bott. One can equivalently, through Serre's GAGA, view this as a result in complex algebraic geometry in the Zariski topology. Formulation Let be a semisimple Lie group or algebraic group over , and fix a maximal torus along with a Borel subgroup which contains . Let be an integral weight of ; defines in a natural way a one-dimensional representation of , by pulling back the representation on , where is the unipotent radical of . Since we can think of the projection map as a principal -bundle, for each we get an associated fiber bundle on (note the sign), which is obviously a line bundle. Identifying with its sheaf of holomorphic sections, we consider the sheaf cohomology groups . Since acts on the total space of the bundle by bundle automorphisms, this action naturally gives a -module structure on these groups; and the Borel–Weil–Bott theorem gives an explicit description of these groups as -modules. We first need to describe the Weyl group action centered at . For any integral weight and in the Weyl group , we set , where denotes the half-sum of positive roots of . It is straightforward to check that this defines a group action, although this action is not linear, unlike the usual Weyl group action. Also, a weight is said to be dominant if for all simple roots . Let denote the length function on . Given an integral weight , one of two cases occur: There is no such that is dominant, equivalently, there exists a nonidentity such that ; or There is a unique such that is dominant. The theorem states that in the first case, we have for all ; and in the second case, we have for all , while is the dual of the irreducible highest-weight representation of with highest weight . It is worth noting that case (1) above occurs if and only if for some positive