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 BorelBottWeilTheorem 2013-03-22 13:50:52 2013-03-22 13:50:52 mathcam (2727) mathcam (2727) 7 mathcam (2727) Theorem msc 14M15