Borel-Bott-Weil theorem

Let G be a semisimple Lie group, and λ be an integral weight for that group. λ naturally defines a one-dimensional representationPlanetmathPlanetmath Cλ of the Borel subgroup B of G, by simply pulling back the representation on the maximal torus T=B/U where U is the unipotent radicalPlanetmathPlanetmath of G. Since we can think of the projection map π:GG/B as a principle B-bundle (, to each Cλ, we get an associated fiber bundleMathworldPlanetmath λ on G/B, which is obviously a line bundleMathworldPlanetmath. Identifying λ with its sheaf of holomorphic sectionsPlanetmathPlanetmathPlanetmathPlanetmath, we consider the sheaf cohomology groups Hi(λ). Realizing 𝔤, the Lie algebraMathworldPlanetmath of G, as vector fields on G/B, we see that 𝔤 acts on the sections of λ over any open set, and so we get an action on cohomology groupsPlanetmathPlanetmath. This integrates to an action of G, which on H0(λ) is simply the obvious action of the group.

The Borel-Bott-Weil theorem states the following: if (λ+ρ,α)=0 for any simple rootMathworldPlanetmath α of 𝔤, then


for all i, where ρ is half the sum of all the positive roots. Otherwise, let wW, the Weyl groupMathworldPlanetmathPlanetmath of G, be the unique element such that w(λ+ρ) is dominant (i.e. (w(λ+ρ),α)>0 for all simple roots α). Then


where Vλ is the unique irreducible representation of highest weight λ, and Hi(λ)=0 for all other i. In particular, if λ is already dominant, then Γ(λ)Vλ, and the higher cohomologyPlanetmathPlanetmath of λ vanishes.

If λ is dominant, than λ is generated by global sections, and thus determines a map


This map is an obvious one, which takes the coset of B to the highest weight vector v0 of Vλ. This can be extended by equivariance since B fixes v0. This provides an alternate description of λ.

For example, consider G=SL2. G/B is P1, the Riemann sphere, and an integral weight is specified simply by an integer n, and ρ=1. The line bundle n is simply 𝒪(n), whose sections are the homogeneous polynomials of degree n. This gives us in one stroke the representation theory of SL2: Γ(𝒪(1)) is the standard representation, and Γ(𝒪(n)) is its nth symmetric power. We even have a unified decription of the action of the Lie algebra, derived from its realization as vector fields on P1: if H,X,Y are the standard generatorsPlanetmathPlanetmathPlanetmath of 𝔰𝔩2, then

H =xddx-yddy
X =xddy
Y =yddx
Title Borel-Bott-Weil theorem
Canonical name BorelBottWeilTheorem
Date of creation 2013-03-22 13:50:52
Last modified on 2013-03-22 13:50:52
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Theorem
Classification msc 14M15