Borel–Bott–Weil theorem

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In mathematics, the Borel–Bott–Weil 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 built on an earlier theorem of Armand Borel and Andre Weil, dealing just with the section case, the extension being provided by Raoul Bott.

Let G be a semisimple Lie group, and λ be an integral weight for that group; λ defines in a natural way a one-dimensional representation C_λ of the Borel subgroup B of G, by pulling back the representation on the maximal torus T = B/U, where U is the unipotent radical of B. Since we can think of the projection map G → G/B as a principal B-bundle, for each C_λ we get an associated fiber bundle L_λ on G/B, which is obviously a line bundle. Identifying L_λ with its sheaf of holomorphic sections, we consider the sheaf cohomology groups Hⁱ(L_λ). Realizing g, the Lie algebra of G, as vector fields on G/B, we see that g acts on the sections of any open set, and so we get an action on cohomology groups. This integrates to an action of G, which on H⁰(L_λ)is simply the evident action of the group.

The Borel–Bott–Weil theorem states the following: if

(λ + ρ,α) = 0

for any simple root α of g, then

Hⁱ(L_λ) = 0 for all i

where ρ is half the sum of all the positive roots. Otherwise, let w in W, the Weyl group 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

H^l(w)(L_λ)

is equivalent to V_λ,the unique irreducible representation of highest weight λ, and

Hⁱ(L_λ) = 0

for all other i. In particular, if λ is already dominant, then

Γ(L_λ) is equivalent to V_λ,

and the higher cohomology of L_λ vanishes.

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

G/B → P(Γ(L_λ).

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

[edit] Example

For example, consider G = SL₂(C), for which G/B is the Riemann sphere, an integral weight is specified simply by an integer n, and ρ = 1. The line bundle L_n is O(n), whose sections are the homogeneous polynomials of degree n (i.e. the binary forms). This gives us at a stroke the representation theory of g: Γ(O(1)) is the standard representation, and Γ(O(n)) is its n-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 H, X, Y are the standard generators of sl₂(C), then we can write