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Given a closed (possibly singular) projective variety $V$ with a symplectic structure and a torus action, there is a moment map $\mu: V \rightarrow Lie(T)^*$. Note that the dimension of $T$ could be much smaller than the dimension of $V$.

How much can we say about the fibers of this moment map $\mu$? Any references?

I am most interested in the case where the variety $V$ is an MV cycle in an affine Grassmannian for an algebraic group $G$. The maximal torus $T \subset G$ acts on the affine Grassmannian. A $T-$equivariant moment map $\mu$ would send an MV cycle to the corresponding MV polytope. What are the fibers of $\mu$ in this case?

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  • $\begingroup$ What one's supposed to do with those fibers is quotient them by $T$, obtaining projective varieties. If you do this at a regular value, the action has finite stabilizers. If you're very lucky, they're trivial, and you have a principal bundle, and the question becomes "which $T$-bundle is this over the quotient", which is answered by its $Lie(T)^*$-valued first Chern class $c$. This was exactly the question Duistermaat-Heckman felt they were answering in "On the Variation in the Cohomology of the Symplectic Form of the Reduced Phase Space"; $c=$ the derivative of $\omega_{red}$ on the base. $\endgroup$
    – Allen Knutson
    Commented Nov 21, 2015 at 21:58

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