Atiyah-Guillemin-Sternberg Convexity Theorem

Theorem5.1Atiyah-Guillemin-Sternberg Convexity Theorem

Let $(M^{2n}, \omega)$ be a compact connected symplectic manifold with a Hamiltonian action of a torus $T^k$ and moment map $\mu: M \to \mathbb{R}^k$ . Then $\mu(M)$ is a convex polytope, equal to the convex hull of the images of the $T$ -fixed points: $\mu(M) = \mathrm{Conv}\{\mu(p) : p \in M^T\}$ .

Proof

We sketch the proof for the case of a Hamiltonian $S^1$ -action. The general case follows by applying the $S^1$ result to each circle subgroup.

Connectedness of fibers: The key step is showing that $\mu^{-1}(c)$ is connected for every $c$ in the image. This follows from Morse-theoretic arguments: $\mu$ is a Morse-Bott function whose critical sets are the $S^1$ -fixed submanifolds. The indices of these critical sets are all even (since the normal bundles are complex), so the level sets remain connected as $c$ varies.

Convexity: Since fibers are connected and $\mu$ is proper (as $M$ is compact), the image $\mu(M) \subseteq \mathbb{R}$ is a closed interval $[\min \mu, \max \mu]$ , which is convex. For $T^k$ , one shows that the image is the intersection of half-spaces determined by projections to each circle subgroup. $\square$

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ExampleCoadjoint Orbit of $SU(3)$

For a coadjoint orbit $\mathcal{O}$ of $SU(3)$ through a regular element, the moment map for the maximal torus $T^2 \subset SU(3)$ has image equal to a hexagon (the convex hull of the Weyl group orbit of the highest weight). This is a fundamental example in representation theory.

RemarkNon-Abelian Convexity

Kirwan's convexity theorem generalizes to non-abelian groups: for a Hamiltonian $G$ -action, the intersection $\mu(M) \cap \mathfrak{t}^*_+$ (with a positive Weyl chamber) is a convex polytope. This refines the abelian result by incorporating the full symmetry structure.