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
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
) 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 of L over Uα. Then we can specify a section s by
the collection of functions A(Uα) given by

= · 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) · 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 by the condition (sα, sα) = 1, this condition becomes
θα = −θα.
Let cα,
be the transition functions, so that = 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
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