Non-abelian convexity by symplectic cuts

Authors: Eugene Lerman, Eckhard Meinrenken, Sue Tolman, Chris Woodward

Year: 1996

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Literature Notes

Section 3 - The principal cross-section

Theorem 3.7: Principal wall + cross-section

Let be a compact connected Lie group, and a connected Hamiltonian G-space with moment map .

1. There exists a unique open wall of the Weyl chamber with the property that is dense in .
2. The preimage is a connected symplectic -invariant suborbifold of , and the restriction of to is a moment map for the action of the maximal torus . (I really only care that it is connected).
3. The saturation is dense in .

In this case is called the principal wall and the principal cross-section.

Lemma 3.8: Affine subspace for

This lemma is used to prove existence of the principal wall . It is used since is dense in so we can use this to look at most of the image set (then the rest is in the closure).

Let be a compact, connected Lie group, and a connected Hamiltonian -space (orbifold) with moment map .

1. For all (where denotes the principal orbit type stratum for the action on ), the isotropy Lie algebra is an ideal in , i.e.

2. All points in the intersection have the same isotropy Lie algebra .
3. Given let be the affine subspace , where is the annihilator of in . The intersection is a connected, relatively open subset of .

Proof of Lemma 3.8

Lemma 3.9: Other walls other than principal wall

Let be an open wall which is not the principal wall . Then the intersection is either empty or is a submanifold of codimension at least 3 in .

Proof of Lemma 3.9

Proof of Theorem 3.7

Since the affine subspace from Lemma 3.8 is affine, and is made from half-planes dividing up , then there is a lowest dimensional wall such that

this also follows from the fact that is a stratified topological space, so thus, it follows the frontier condition, so there is a “lowest” stratum with this property.

Since the moment map is continuous, and , then

By Lemma 3.8,

\text{span}(\sigma) \simeq \mathfrak z(\mathfrak g_\sigma) \simeq [\mathfrak g_\sigma, \mathfrak g_\sigma]^\text{ann} \cap \mathfrak g_\sigma^*

\newcommand{\g}{\mathfrak g}

\mathfrak h^\text{ann} \cap \mathfrak{g}\sigma^* \subset [\g\sigma, \g_\sigma]^\text{ann} \cap \g_\sigma^* \implies [\g_\sigma, \g_\sigma] \subset \mathfrak h

M_\text{prin} \backslash (G \cdot Y \cap M_\text{prin}) = \bigcup_{\tau \subset \overline \sigma} \mu^{-1}(G \cdot \tau) \cap M_\text{prin}