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 $\pi\colon G\to G/B$ as a principle $B$ -bundle (http://planetmath.org/PrincipleBundle), to each $C_{\lambda}$ , we get an associated fiber bundle $\mathcal{L}_{\lambda}$ on $G/B$ , which is obviously a line bundle. Identifying $\mathcal{L}_{\lambda}$ with its sheaf of holomorphic sections, we consider the sheaf cohomology groups $H^{i}(\mathcal{L}_{\lambda})$ . Realizing $\mathfrak{g}$ , the Lie algebra of $G$ , as vector fields on $G/B$ , we see that $\mathfrak{g}$ acts on the sections of $\mathcal{L}_{\lambda}$ over any open set, and so we get an action on cohomology groups. This integrates to an action of $G$ , which on $H^{0}(\mathcal{L}_{\lambda})$ 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 $\mathfrak{g}$ , then

H^{i}(\mathcal{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^{\ell(w)}(\mathcal{L}_{\lambda})\cong V_{\lambda}

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

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

m_{\lambda}\colon G/B\to\mathbb{P}\left(\Gamma(\mathcal{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 $\mathcal{L}_{\lambda}$ .

For example, consider $G=\mathrm{SL}_{2}\mathbb{C}$ . $G/B$ is $\mathbb{C}P^{1}$ , the Riemann sphere, and an integral weight is specified simply by an integer $n$ , and $\rho=1$ . The line bundle $\mathcal{L}_{n}$ is simply $\mathcal{O}(n)$ , whose sections are the homogeneous polynomials of degree $n$ . This gives us in one stroke the representation theory of $\mathrm{SL}_{2}\mathbb{C}$ : $\Gamma(\mathcal{O}(1))$ is the standard representation, and $\Gamma(\mathcal{O}(n))$ 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 $\mathbb{C}P^{1}$ : if $H, X, Y$ are the standard generators of $\mathfrak{sl}_{2}\mathbb{C}$ , then

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

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