§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.
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