PlanetMath: Borel-Bott-Weil theorem

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Borel-Bott-Weil theorem

(Theorem)

Let 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 of , by simply pulling back the representation on the maximal torus where is the unipotent radical of . Since we can think of the projection map $\pi\colon G\to G/B$ as a principle -bundle, to each $C_\lambda$ , we get an associated fiber bundle $\L _\lambda$ on , 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 $\mathfrak{g}$ , the Lie algebra of , as vector fields on , we see that $\mathfrak{g}$ 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 , 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 $\mathfrak{g}$ , then

$\displaystyle H^i(\L _\lambda)=0$

for all

, where $\rho$ is half the sum of all the positive roots. Otherwise, let $w\in W$ , the Weyl group of

, be the unique element such that $w(\lambda+\rho)$ is dominant (i.e. $(w(\lambda+\rho),\alpha)>0$ for all simple roots $\alpha$ ). Then

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

where $V_\lambda$ is the unique irreducible representation of highest weight $\lambda$ , and $H^i(\L _\lambda)=0$ for all other

. In particular, if $\lambda$ is already dominant, then $\Gamma(\L _\lambda)\cong V_\lambda$ , 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

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

This map is an obvious one, which takes the coset of

to the highest weight vector

of $V_\lambda$ . This can be extended by equivariance since

fixes

. This provides an alternate description of $\L _\lambda$ .

For example, consider $G=\mathrm{SL}_2\mathbb{C}$ . is $\mathbb{C}P^1$ , the Riemann sphere, and an integral weight is specified simply by an integer , and $\rho=1$ . The line bundle $\L _n$ is simply $\O (n)$ , whose sections are the homogeneous polynomials of degree . This gives us in one stroke the representation theory of $\mathrm{SL}_2\mathbb{C}$ : $\Gamma(\O (1))$ is the standard representation, and $\Gamma(\O (n))$ is its 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 are the standard generators of $\mathfrak{sl}_2\mathbb{C}$ , then

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

"Borel-Bott-Weil theorem" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Cross-references: generators, even, symmetric power, theory, degree, homogeneous polynomials, integer, Riemann sphere, vector, coset, map, global sections, generated by, vanishes, cohomology, highest weight, irreducible, dominant, Weyl group, positive roots, sum, simple root, obvious, cohomology groups, action, open set, acts on, vector fields, Lie algebra, sheaf cohomology, sections, holomorphic, sheaf, line bundle, fiber bundle, projection map, radical, maximal torus, Borel subgroup, representation, group, integral weight, Lie group, semisimple
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This is version 4 of Borel-Bott-Weil theorem, born on 2003-08-14, modified 2004-03-20.
Object id is 4585, canonical name is BorelBottWeilTheorem.
Accessed 3695 times total.

Classification:

AMS MSC:

14M15 (Algebraic geometry :: Special varieties :: Grassmannians, Schubert varieties, flag manifolds)

Pending Errata and Addenda

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