§2. Symplectic structure on coadjoint orbits 11
Proof. The proof follows from Theorem 3 and the explicit formulae above
for θ,
X∗, X∗.
For the left action the moment map looks especially simple (because we
are using the left trivialization): μ (g, F ) = F .
The fiber of the moment map over F
g∗
is G × {F }. So, the reduced
manifold is G/Stab(F ) ΩF . We omit the (rather tautological) verification
of the equality σ0 = σΩ.
We could get the same result using the right action of G on T
∗G.
Indeed,
here μ (g, F ) = K(g)F and
μ−1(ΩF
) = G × ΩF . Therefore, the reduced
manifold is (G × ΩF )/G ΩF .
2.4. Integrality condition.
In Appendix II.2.4 we explain how to integrate a differential k-form on
M over an oriented smooth k-dimensional submanifold.
More generally, define a real (resp. integral) singular k-cycle in a
manifold M as a linear combination C =

i
ci · ϕi(Mi) of images of smooth
k-manifolds Mi under smooth maps ϕi : Mi M with real (resp. integer)
coefficients ci. Then we can define the integral of the form ω Ωk(M) over
a singular k-cycle C as
C
ω =
i
ci ·
Mi
ϕi
∗(ω)
.
It is known that the integral of a closed k-form over a singular k-
dimensional cycle C depends only on the homology class [C]
Hsing(M).4
Moreover, a k-form is exact iff its integral over any k-cycle vanishes.
For future use we make the following
Definition 2. A coadjoint orbit Ω is integral if the canonical form σ has
the property
(16)
C
σ Z for every integral singular 2-cycle C in Ω.
In particular, it is true when C is any smooth 2-dimensional submanifold
S Ω.
4We
will not give the accurate definitions here of the groups Hsing(M, R) and Hsing(M, Z) of
singular homologies of a manifold M. For the applications in representation theory the information
given in this section is quite enough.
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