10 1. THE CLASSICAL THEORY: PART I

In general, given

• a homogeneous space

Y = A/B

where A is a Lie group and B ⊂ A is a closed subgroup, and

• a linear representation r : B → Aut E where E is a complex vector space,

there is an associated homogeneous vector bundle

E := A ×B E

Y = A/B

where A×B E is the trivial vector bundle A×E factored by the equivalence relation

(a, e) ∼ (ab,

r(b−1)e)

where a ∈ A, e ∈ E, b ∈ B. The group A acts on E by a·(a , e) = (aa , e) and there

is an A-equivariant action on E → Y . There is an evident notion of a morphism of

homogeneous vector bundles; then E → Y is trivial as a homogeneous vector bundle

if, and only if, r : B → Aut(E) is the restriction to B of a representation of A.

Example. Let τ0 ∈ H ⊂

P1

be the reference point. For the standard linear

representation of SL2(C) on VC, the Borel subgroup B is the stability group of the

flag

(0) ⊂

Vτ1,0

0

⊂ VC.

It follows that there is over

P1

an exact sequence of SL2(C)-homogeneous vector

bundles

0 → OP1 (−1) → V → OP1 (1) → 0

where V =

P1

× VC with g ∈ SL2(C) acting on V by g · ([z],v) = ([gz],gv). The

restriction to H of this sequence is an exact sequence of SL2(R)-homogeneous bun-

dles

0 →

V1,0

→ V →

V0,1

→ 0.

The bundle V1,0 is given by the representation

cos θ − sin θ

sin θ cos θ

→

eiθ

of SO(2). Using the form Q the quotient bundle

V/V1,0

:=

V0,1

is identified with

the dual bundle

V1,0∗.

The canonical line bundle is

ωP1

∼

= OP1 (−2).

Thus

ωH

∼

=

(V1,0)⊗2.

Proof. For the Grassmannian Y = Gr(n, E) of n-planes P in a vector space E

there is the standard GL(E)-equivariant isomorphism

TP Y

∼

=

Hom(P, E/P ).

In the case above where E = C2 and z = [

z0

z1

] ∈ P1 we have

TzP1

∼

=

Vz1,0∗

⊗

VC/Vz1,0