§2. Symplectic structure on coadjoint orbits 13

form of the connection ∇ and is denoted curv ∇. (In some books another

normalization is used and the real form

1

2πi

Θ is called the curvature form.)

Exercise 2. Prove that

(curv ∇)(v, w) = [∇v, ∇w] − ∇[v, w].

Hint. Use the formula for ∇v above and the formula for dθ. ♣

We will not bother the reader with the verification of the following fact:

Lemma 5. The

ˇ

Cech cocycle corresponding to the 2-form curv ∇ has the

form

cα,β,γ = log cα,

β

+ log cβ,

γ

+ log cγ, α.

From this formula and from the relation cα,

β

· cβ,

γ

· cγ,

α

= 1 it follows

immediately that the cohomology class of curv ∇ belongs to 2πi ·

H2(Ω,

Z).

Hence, the form σ =

1

2πi

curv ∇ belongs to an integral cohomology class.

Conversely, let σ be a real 2-form on Ω, and let θα be the real antideriv-

ative of σ on Uα. Then we can define the functions cα,β on Uα,β so that

dcα,β = 2πi(θβ − θα). If σ has the property [σ] ∈

H2(Ω,

Z), these functions

satisfy cα,

β

· cβ,

γ

· cγ,

α

= 1. Hence, they can be considered as the transition

functions of some complex line bundle L over Ω. Since the θα are real, we

can assume that |cα, β| = 1. Therefore, L admits a scalar product in fibers

such that ∇v = v + 2πiθα(v) is a Hermitian connection on L.

(ii)⇐⇒(iii). Since stab(F ) = ker BF , the representation (18) is in fact a

representation of an abelian Lie group A =

Stab◦(F )/[Stab◦(F

),

Stab◦(F

)].

This group has the form

Tk

×

Rl.

The representation (18) corresponds to a

unitary 1-dimensional representation of A iff it takes integer values on the

k-dimensional lattice Λ =

exp−1(e).

Let X ∈ Λ and denote by γ the loop in

Stab◦(F

) which is the image of

the segment [0, X] ⊂ stab(F ) under the exponential map.

Since G is supposed to be simply connected, the loop γ is the boundary of

some 2-dimensional surface S in G, whose projection to Ω we denote by p(S).

The correspondence [γ] [S] is precisely the isomorphism π1(Stab(F ))

π2(Ω) (see formula (6) in Appendix I.2.3). Finally we have

F, X =

γ

θF =

S

dθF =

p(S)

σ.