1
$\begingroup$

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?

$\endgroup$
4
$\begingroup$

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).

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Han Jin Ma Sep 9 '15 at 16:10
  • $\begingroup$ For $SU(3)$, does the 2-dim space of symplectic forms imply another equivariant Hermitian metric? $\endgroup$ – Han Jin Ma Sep 9 '15 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. $\endgroup$ – Allen Knutson Sep 10 '15 at 1:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.