Borel-Bott-Weil theorem

Borel-Bott-Weil theorem

Let

G

be a

semisimple Lie group

, and

λ

be an

integral weight

for that group.

λ

naturally defines a one-dimensional

representation

C

λ

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

π

:

G

→

G

/

B

as a principle

B

-bundle (

http://planetmath.org/PrincipleBundle

), to each

C

λ

, we get an associated

fiber bundle

ℒ

λ

on

G

/

B

, which is obviously a

line bundle

. Identifying

ℒ

λ

with its sheaf of holomorphic

sections

, we consider the

sheaf cohomology

groups

H

i

⁢

(

ℒ

λ

)

. Realizing

𝔤

, the

Lie algebra

of

G

, as vector fields on

G

/

B

, we see that

𝔤

acts on the sections of

ℒ

λ

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

λ

)

is simply the obvious action of the group.

The

Borel-Bott-Weil theorem

states the following: if

(

λ

+

ρ

,

α

)

=

0

for any

simple root

α

of

𝔤

, then

H

i

⁢

(

ℒ

λ

)

=

0

for all

i

, where

ρ

is half the sum of all the

positive roots

. Otherwise, let

w

∈

W

, the

Weyl group

of

G

, be the unique element such that

w

⁢

(

λ

+

ρ

)

is dominant (i.e.

(

w

⁢

(

λ

+

ρ

)

,

α

)

>

0

for all simple roots

α

).
Then

H

ℓ

⁢

(

w

)

⁢

(

ℒ

λ

)

≅

V

λ

where

V

λ

is the unique

irreducible representation

of

highest weight

λ

, and

H

i

⁢

(

ℒ

λ

)

=

0

for all other

i

. In particular, if

λ

is already dominant, then

Γ

⁢

(

ℒ

λ

)

≅

V

λ

, and the higher

cohomology

of

ℒ

λ

vanishes.

If

λ

is dominant, than

ℒ

λ

is generated by global sections, and thus determines a map

m

λ

:

G

/

B

→

ℙ

⁢

(

Γ

⁢

(

ℒ

λ

)

)

.

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

B

to the highest weight vector

v

0

of

V

λ

. This can be extended by equivariance since

B

fixes

v

0

. This provides an alternate description of

ℒ

λ

.

For example, consider

G

=

SL

2

⁢

ℂ

.

G

/

B

is

ℂ

⁢

P

1

, the Riemann sphere, and an integral weight is specified simply by an

integer

n

, and

ρ

=

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

SL

2

⁢

ℂ

:

Γ

⁢

(

𝒪

⁢

(

1

)

)

is the

standard representation

, and

Γ

⁢

(

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

ℂ

⁢

P

1

: if

H

,

X

,

Y

are the standard

generators

of

𝔰

⁢

𝔩

2

⁢

ℂ

, then

H

=

x

⁢

d

d

⁢

x

-

y

⁢

d

d

⁢

y

X

=

x

⁢

d

d

⁢

y

Y

=

y

⁢

d

d

⁢

x

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

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