§2. Symplectic structure on coadjoint orbits 9
2.3.∗
The third (symplectic reduction) approach.
We apply the symplectic reduction procedure described in Appendix
II.3.2 to the special case when the symplectic manifold M is the cotangent
bundle T
∗G
over a Lie group G. This case has its peculiarities.
First, the bundle T
∗G
is trivial. Namely, we shall use the left action of
G on itself to make the identification T
∗(G)
G ×
g∗.
In matrix notation
the covector g · F Tg−1

G corresponds to the pair (g, F ) G ×
g∗.3
Further, the set T
∗(G)
is itself a group with respect to the law
(g1, F1)(g2, F2) = (g1g2,
K(g2)−1F1
+ F2).
If we identify (g, 0) T
∗(G)
with g G and (e, F ) T
∗(G)
with F
g∗,
then T
∗(G)
becomes a semidirect product G
g∗.
So, we can write (g, F ) =
g · F both in matrix notation and in the sense of the group law in T
∗G.
Note also the identity g · F ·
g−1
= K(g)F .
Since T
∗G
is a Lie group, the tangent bundle T (T
∗(G))
is also trivial.
We identify T(g,F )T
∗G
with Lie (T
∗G) g⊕g∗
using the left shift and obtain
T (T
∗(G))
(G ×
g∗)
× (g
g∗).
Theorem 3. The canonical symplectic structure on T
∗(G)
in the trivial-
ization above is given by the bilinear form σ:
(12) σ(g,F )(X1 F1, X2 F2) = F1, X2 F2, X1 F, [X1, X2] .
Proof. Let us compute first the canonical 1-form θ on T
∗(G).
For a tangent
vector v = (g, F ; X, F ) T(g,F )T
∗G
the projection to TgG equals X ·
g−1.
Therefore, θ(v) = g · F, X ·
g−1
= F, X .
Now we can compute σ as the exterior derivative of θ:
σ(ξ1, ξ2) = ξ1θ(ξ2) ξ2θ(ξ1) θ([ξ1, ξ2])
for any vector fields ξ1, ξ2 on T
∗G.
We choose ξ1, ξ2 as the left-invariant fields on the Lie group T
∗G
with
initial values
ξ1(e, 0) = (X1, F1), ξ2(e, 0) = (X2, F2).
3Recall that g · F is defined by g · F, ξ = F, ξ · g−1 .
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