Projections of orbifolds

Question

A while back I came across orbifolds, in particular the quotients $SU(2)/U(1)\cong S^2$, $SU(3)/(SU(2)\times U(1)\cong \mathbb{C}P^2$ and $SU(3)/(U(1)\times U(1))$. The way I needed them, was as an embedding in the corresponding Lie-algebra $\mathfrak{su}(n)$.

To be a bit more precise: Consider first the $SU(2)$ case. Take some $X\in \mathfrak{h}\subseteq\mathfrak{su}(2)$. Then the orbit generated by the adjoint action is of course $\{Ad(g)X\ |\ g\in SU(2)\}\cong S^2$. For $SU(3)$ there are two choices for some initial "seed", such that the adjoint action generates either $\mathbb{C}P^2$ or $SU(3)/(U(1)\times U(1))$.

Now let $\Pi_{\mathfrak{h}}:\mathfrak{su}(n)\rightarrow\mathfrak{h}$ be the projection onto the Cartan sub-algebra. Not surprisingly, the image $\Pi_\mathfrak{h}(S^2)$ is an interval [-1,1] -- up to some scaling-factor -- in the one-dimensional subspace $\mathfrak{h}$. For $SU(3)$, this becomes more interesting. I noticed that the projection $\Pi_{\mathfrak{h}}(\mathbb{C}P^2)$ is an equilateral triangle and the projection $\Pi_{\mathfrak{h}}(SU(3)/(U(1)\times U(1)))$ is a (not necessarily regular) hexagon.

Coincidently, these are just the two geometric forms one obtains for the weight diagrams of the irreducible representations $D(N,0)$ and $D(0,N)$, or $D(N,M)$ with $N\not=0\not=M$. This is obviously true for the sphere as well, since the weights are labeled by $m=-l,\dots,l$, $l\in\frac{1}{2}\mathbb{N}$ in corresponce to the interval obtained from the projection $\Pi_{\mathfrak{h}}$.

This seems to be to much of a coincidence, but I'm pretty sure this is already known. However, so far I haven't found anything to confirm myself that such a correspondence holds for any quotient of $SU(n)$ (possibly in some sense also for other Lie-groups). Does anyone know any sources, where I can find a solution to this puzzle?

Answer 1

Yes, it is already known.

The story perhaps starts with:

Kostant's convexity theorem. Let $G/K$ be a symmetric space of compact type, $\mathfrak g=\mathfrak k +\mathfrak p$ the decomposition into the eigenspaces of the involution, $\mathfrak a$ a maximal Abelian subspace of $\mathfrak p$ and $W$ the restricted Weyl group, namely, the normalizer of $\mathfrak a$ in $K$ modulo its centralizer. Then, for any $a\in\mathfrak a$, the orthogonal projection of the adjoint orbit $\mathrm{Ad}_K(a)$ to $\mathfrak a$ (with respect to an $\mathrm{Ad}_K$-invariant inner product) is the convex polytope whose vertices are exactly the $w(a)$ for $w\in W$.

In particular, this theorem includes the case of adjoint or coadjoint orbits of a compact connected semisimple Lie group $G$, since $G$ with a bi-invariant Riemannian metric is the symmetric space $G\times G/\Delta_G$, where $\Delta_G$ is the diagonal subgroup. Coadjoint orbits $M$ are symplectic manifolds with respect to the Kirillov-Kostant-Souriau form, and the orthogonal projection onto the Lie algebra of the maximal torus $T$ is exactly the moment map of the $T$-action on $M$. A symplectic version of Kostant's theorem is then:

Atiyah-Guillemin-Sternberg's convexity theorem. Let $M$ be a compact symplectic manifold with a Hamiltonian action of a torus $T$. Then the image of the moment map $\mu:M\to\mathfrak t^*$ is the convex hull of the image under $\mu$ of the fixed point set of $T$ on $M$.

Orbits of isotropy representations of a symmetric spaces are isoparametric submanifolds of Euclidean space, namely, the eigenvalues of the shape operator along a locally defined parallel normal field are constant, and the normal bundle is flat. This provides for another generalization of Kostant's theorem:

Terng's convexity theorem. Let $M$ be a compact full isoparametric submanifold of Euclidean space and fix $p\in M$. Then the orthogonal projection of $M$ (or any parallel manifold through, say, $q\in p+\nu_pM$) to the normal space $p+\nu_pM$ is the convex hull of orbit of $p$ (resp. $q$) under the Weyl group $W$ associated to $M$.

Reference: Convexity theorem for isoparametric submanifolds. Ch. L. Terng. Inventiones mathematicae 85 (1986), 487-492. (article)