Unique Equivariant Symplectic Structure for the Full Flag Manifold of $SU(3)$?

I was looking at the following interesting question about the number of equivariant almost complex structures on the full flag manifold of $SU(3)$, and I began to wonder how many equivariant symplectic structures there are on the same space, i.e. $M := SU(3)/T^2$. Now as I understand it, there is a unique equivariant K\"ahler metric for $M$, which will of course give us one equivariant symplectic structure, and since symplectic structures which do not arise from K\"ahler metrics are hard to come by, I would conjecture that this is the only equivariant symplectic structure for this space. Is this true?

1 Answer

If $G$ acts on $M$ (both compact and finite-dimensional) preserving the symplectic form, and $M$ is simply-connected, the action is Hamiltonian. Then $M$ maps symplectomorphically to a coadjoint orbit of $G$. This gives a $rank(G)$-dimensional family of symplectic forms, not a unique one.

You should consider the $SU(2)$ case first, where $\mathbb{CP}^1$ doesn't have a unique area form (or area).

• I guess there is some misunderstanding here, when I said unique, I intended "unique up to scalar multiple". I thought this was obvious . . . but obviously not. Apologies for my sloppiness. – Han Jin Ma Sep 9 '15 at 16:10
• For $SU(3)$, does the 2-dim space of symplectic forms imply another equivariant Hermitian metric? – Han Jin Ma Sep 9 '15 at 16:12
• Yes, there is a $[0,1]$ interval's worth of invariant (I don't know why you say equivariant) Hermitian metrics, considered up to scale. – Allen Knutson Sep 10 '15 at 1:56