§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.