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?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
3
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).
answered Sep 9, 2015 at 11:24
-
$\begingroup$ 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. $\endgroup$ Commented Sep 9, 2015 at 16:10
-
$\begingroup$ For $SU(3)$, does the 2-dim space of symplectic forms imply another equivariant Hermitian metric? $\endgroup$ Commented Sep 9, 2015 at 16:12
-
$\begingroup$ 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. $\endgroup$ Commented Sep 10, 2015 at 1:56