Borel-Bott-Weil theorem

Let $G$ be a semisimple Lie group, and $\lambda$ be an integral weight for that group. $\lambda$ naturally defines a one-dimensional representation $C_{\lambda }$ 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 radical of $G$. Since we can think of the projection map $\pi :G\to G/B$ as a principle $B$-bundle (http://planetmath.org/PrincipleBundle), to each $C_{\lambda }$, we get an associated fiber bundle ${ℒ}_{\lambda }$ on $G/B$, which is obviously a line bundle. Identifying ${ℒ}_{\lambda }$ with its sheaf of holomorphic sections, we consider the sheaf cohomology groups $H^i({ℒ}_{\lambda })$. Realizing $𝔤$, the Lie algebra of $G$, as vector fields on $G/B$, we see that $𝔤$ acts on the sections of ${ℒ}_{\lambda }$ over any open set, and so we get an action on cohomology groups. This integrates to an action of $G$, which on $H^0({ℒ}_{\lambda })$ is simply the obvious action of the group.

The Borel-Bott-Weil theorem states the following: if $(\lambda +\rho ,\alpha )=0$ for any simple root $\alpha$ of $𝔤$, then

$$H^i({ℒ}_{\lambda })=0$$

for all $i$, where $\rho$ 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 $w(\lambda +\rho )$ is dominant (i.e. $(w(\lambda +\rho ),\alpha )>0$ for all simple roots $\alpha$). Then

$$H^{\mathrm{\ell }(w)}({ℒ}_{\lambda })\cong V_{\lambda }$$

where $V_{\lambda }$ is the unique irreducible representation of highest weight $\lambda$, and $H^i({ℒ}_{\lambda })=0$ for all other $i$. In particular, if $\lambda$ is already dominant, then $\mathrm{\Gamma }({ℒ}_{\lambda })\cong V_{\lambda }$, and the higher cohomology of ${ℒ}_{\lambda }$ vanishes.

If $\lambda$ is dominant, than ${ℒ}_{\lambda }$ is generated by global sections, and thus determines a map

$$m_{\lambda }:G/B\to \mathbb{P}\left(\mathrm{\Gamma }({ℒ}_{\lambda })\right).$$

This map is an obvious one, which takes the coset of $B$ to the highest weight vector $v_0$ of $V_{\lambda }$. This can be extended by equivariance since $B$ fixes $v_0$. This provides an alternate description of ${ℒ}_{\lambda }$.

For example, consider $G={\mathrm{SL}}_2ℂ$. $G/B$ is $ℂP^1$, the Riemann sphere, and an integral weight is specified simply by an integer $n$, and $\rho =1$. The line bundle ${ℒ}_n$ is simply $𝒪(n)$, whose sections are the homogeneous polynomials of degree $n$. This gives us in one stroke the representation theory of ${\mathrm{SL}}_2ℂ$: $\mathrm{\Gamma }(𝒪(1))$ is the standard representation, and $\mathrm{\Gamma }(𝒪(n))$ is its $n$th symmetric power. We even have a unified decription of the action of the Lie algebra, derived from its realization as vector fields on $ℂP^1$: if $H,X,Y$ are the standard generators of $𝔰{𝔩}_2ℂ$, then

$H$ $=x{\displaystyle \frac{d}{dx}}-y{\displaystyle \frac{d}{dy}}$

$X$ $=x{\displaystyle \frac{d}{dy}}$

$Y$ $=y{\displaystyle \frac{d}{dx}}$

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