12 1. Geometry of Coadjoint Orbits

The integrality condition has important geometric and representation-

theoretic interpretations. They are revealed by the following

Proposition 2. Assume that G is a simply connected Lie group. The fol-

lowing are equivalent:

(i) Ω ⊂

g∗

is integral.

(ii) There exists a G-equivariant complex line bundle over Ω with a G-

invariant Hermitian connection ∇ such that

(17) curv (∇) = 2πiσ.

(iii) For any F ∈ Ω there exists a unitary 1-dimensional representation

χ of the connected Lie group

Stab◦(F

) such that

(18) χ(exp X) =

e2πi F, X

.

Observe that condition (i) is automatically true for homotopically trivial

(i.e. contractible) orbits. It is also true when the canonical form σ is exact.

Proof. (i)⇐⇒(ii). Let L be a complex line bundle over Ω. Choose a cov-

ering of Ω by open sets {Uα}α∈A such that for any α ∈ A there exists a

non-vanishing section sα of L over Uα. Then we can specify a section s by

the collection of functions fα ∈ A(Uα) given by

s

Uα

= fα · sα.

A connection ∇ in a line bundle L is given by a family of differential

1-forms θα. Namely, define θα by ∇vsα = θα(v) · sα for any v ∈ Vect (Uα).

In terms of these forms the covariant derivative is

∇v = v + θα(v), i.e. s ↔ {fα} ⇒ ∇vs ↔ {vfα + θα(v)fα}.

The connection is Hermitian if a scalar product is defined in all fibers

so that

v · (s1, s2) = (∇vs1, s2) + (s1, ∇vs2).

If we normalize sα by the condition (sα, sα) = 1, this condition becomes

θα = −θα.

Let cα,

β

be the transition functions, so that fα = cα, βfβ on Uα, β. Then

the forms θα satisfy

θβ − θα = d log cα, β.

Therefore, the form dθα coincides with dθβ on Uα, β. Hence, the collection

{dθα} defines a single 2-form Θ on Ω. This form is called the curvature