# 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^{} ${\mathcal{L}}_{\lambda}$ 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 $\U0001d524$, the Lie algebra^{} of $G$, as vector fields on $G/B$, we see that $\U0001d524$ 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 $\U0001d524$, 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

$${H}^{\mathrm{\ell}(w)}({\mathcal{L}}_{\lambda})\cong {V}_{\lambda}$$ |

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 $\mathrm{\Gamma}({\mathcal{L}}_{\lambda})\cong {V}_{\lambda}$, 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

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

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 $G={\mathrm{SL}}_{2}\u2102$. $G/B$ is $\u2102{P}^{1}$, 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 ${\mathrm{SL}}_{2}\u2102$: $\mathrm{\Gamma}(\mathcal{O}(1))$ is the standard representation, and $\mathrm{\Gamma}(\mathcal{O}(n))$ 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 $\u2102{P}^{1}$: if $H,X,Y$ are the standard generators^{} of $\U0001d530{\U0001d529}_{2}\u2102$, 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 |