Borel–Weil–Bott theorem


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 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 , 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:
  1. There is no such that is dominant, equivalently, there exists a nonidentity such that ; or
  2. There is a unique such that is dominant.
The theorem states that in the first case, we have
and in the second case, we have
It is worth noting that case above occurs if and only if for some positive 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 canonical bundle|, whose sections are the homogeneous polynomials of degree . 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 Lie group or for its compact form. Let be a connected complex semisimple Lie group, a Borel subgroup of, and the flag variety. In this scenario, is a complex manifold and a nonsingular algebraic. The flag variety can also be described as a compact homogeneous space, where is a Cartan subgroup of. An integral weight determines a holomorphic line bundle on and the group acts on its space of global sections,
The Borel–Weil theorem states that if is a dominant integral weight then this representation is a holomorphic irreducible highest weight representation of with highest weight. Its restriction to is an irreducible unitary representation of with highest weight, and each irreducible unitary representations of is obtained in this way for a unique value of.

Concrete description

The weight gives rise to a character of the Borel subgroup, which is denoted. Holomorphic sections of the holomorphic line bundle over may be described more concretely as holomorphic maps
for all and.
The action of on these sections is given by
for.

Example

Let be the complex special linear group, with a Borel subgroup consisting of upper triangular matrices with determinant one. Integral weights for may be identified with integers, with dominant weights corresponding to nonnegative integers, and the corresponding characters of have the form
The flag variety may be identified with the complex projective line with homogeneous coordinates and the space of the global sections of the line bundle is identified with the space of homogeneous polynomials of degree on. For, this space has dimension and forms an irreducible representation under the standard action of on the polynomial algebra. Weight vectors are given by monomials
of weights, and the highest weight vector has weight.