§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 .