Example in Guillemin-Sternberg's Convexity Paper

Question

At the end of the introduction of Guillemin-Sternberg's 1982 Inventiones paper "Convexity Properties of the Moment Mapping", there is a nice picture of a non-Abelian moment polytope attributed to Heckman. Is this example worked out somewhere in more detail? Also, are you aware of other illustrations of the intricacies of the non-Abelian convexity theorem?

Answer 1

That particular example is about branching from $SO(5)$ to $SO(4)$. In general, the branching rule from $SO(n)$ to $SO(n-1)$ is multiplicity-free and well-known; it's in, e.g., Zhelobenko's book. So the corresponding nonabelian DH measure on the nonabelian moment polytope will be some $1/N$ multiple of Lebesgue measure.

The "intricacy" they point out there is that the intersection with $t^\ast$ is nonconvex. But basically, intersecting with $t^\ast$ is a silly thing to do: if you push forward Liouville measure along the moment map $M\to g^\ast$, you get a measure on $g^\ast$ that you can't "restrict" to $t^\ast$. Rather, the natural thing is to push forward again along $g^\ast \to t^\ast_+$, dividing by the coadjoint action. And there, the image is indeed convex, by Kirwan's nonabelian convexity theorem.

(That doesn't give the right definition of nonabelian DH measure, though -- you have to divide the result by the volume polynomial that gives the volume of the coadjoint orbits. If $M$ is itself a coadjoint orbit $G\cdot \lambda$, you want to get a Dirac delta at $\lambda$, not that times the volume of $G\cdot \lambda$.)

You asked for another intricacy. One is that the vertices of the moment polytope that lie on the walls of the Weyl chamber are much stickier to interpret than the ones in the interior; there's no simple description of the moment polytope as a convex hull. One hint of this is that even if $M$ is prequantizable, the vertices on Weyl walls may not be at lattice points.