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
