§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.
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