Borel–Bott–Weil theorem

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 André 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 "H^i"("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 "H^0"("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

:"H^i"("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

:"H^i"("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 "v"₀. It can be extended by equivariance since "B" fixes "v"₀. This provides an alternate description of "L"_λ.

Example

For example, consider "G" = "SL"₂("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 "sl"₂("C"), then we can write

:$H\; =\; xfrac\{d\}\{dx\}-yfrac\{d\}\{dy\}$:$X\; =\; xfrac\{d\}\{dy\}$:$Y\; =\; yfrac\{d\}\{dx\}.$

External links

* [http://www-math.mit.edu/~lurie/papers/bwb.pdf A Proof of the Borel-Weil-Bott Theorem] , by Jacob Lurie. Retrieved on Dec. 14, 2007.