§2. Symplectic structure on coadjoint orbits 5

Now we introduce the antisymmetric bilinear form BF on g by the for-

mula

(6) BF (X, Y ) = F, [X, Y ] .

Lemma 1. The kernel of BF is exactly stab(F ).

Proof.

ker BF = X ∈ g BF (X, Y ) = 0 ∀ Y ∈ g

= X ∈ g K∗(X)F, Y = 0 ∀ Y ∈ g

= X ∈ g K∗(X)F = 0 = stab(F ).

Lemma 2. The form BF is invariant under Stab(F ).

Proof.

F, [AdhX, AdhY ] = F, Adh[X, Y ] =

K(h−1)F,

[X, Y ] = F, [X, Y ]

for any h ∈ Stab(F ).

Now we are ready to introduce

Definition 1. Let Ω be a coadjoint orbit in

g∗.

We define the differential

2-form σΩ on Ω by

(7) σΩ(F )(K∗(X)F, K∗(Y )F ) = BF (X, Y ).

The correctness of this definition, as well as the non-degeneracy and G-

invariance of the constructed form follows immediately from the discussion

above.

2. Symplectic structure on coadjoint orbits

The goal of this section is to prove

Theorem 1. The form σΩ is closed, hence defines on Ω a G-invariant

symplectic structure.

There exist several proofs of this theorem that use quite different ap-

proaches. This can be considered as circumstantial evidence of the depth

and importance of the theorem. Three and a half of these proofs are pre-

sented below.