⇤Math diary
May 28, 2023: Morse theory on coadjoint orbits
Let me step back from loop groups a little bit, and talk about good ol’ lie groups. The “orbit method” studies lie groups and their representations using the orbits of the coadjoint action of the lie group $G$ on the dual lie algebra $\mathfrak{g}^\ast$. These coadjoint orbits $\cal O$ carry a canonical symplectic structure, induced from $\mathfrak{g}^\ast$, which has an alternating 2-form built from the lie bracket. Coadjoint orbits are quite well understood, which their geometry often reducing to explicit, computable data of the Lie group.
First off, the coadjoint action of $G$ on $\cal O$ is Hamiltonian, with momentum map $\mu: \cal O \to \mathfrak g^\ast$ simply the inclusion $\cal O \hookrightarrow \mathfrak{g}^\ast$. Choose a maximal torus $T\subset G$, with associated Cartan subalgebra $\mathfrak{h} \subset\mathfrak{g}$. For a generic orbit $\cal O$, $T$ acts effectivly, so we can use the theory of symplectic torus actions. In particular, by the Atiyah-Guilleman-Sternberg convexity theorem, the image of $\cal O$ under the moment map of $T$ is a convex polytope in $\mathfrak{h}$. In fact, the moment map for $T$, which we denote $\mu_T$, is the orthogonal projection of the original moment map onto the lie algebra of $T$:
\[\mu_T = \pi^\perp_\mathfrak {h} \circ \mu\]Since $\mu$ in this case is the inclusion map, this implies Konstant’s convexity theorem: the projection of $\cal O$ to a cartan subalgebra $\mathfrak{h}$ is a convex polytope.
But what is the shape of this polytope?
Theorem
The image of the moment map $\mu_T$ is a convex subset of the Cartan subalgebra $\mathfrak h$. It is the convex hull of an orbit of the Weyl group.
Atiyah-Guilleman-Sternberg’s proof tells us that the image of the moment map is a polytope is in fact the convex hull of the images of the fixed points of the torus action. The fixed points of the action of $T$ on $\mathfrak g ^\ast$ are exactly the associated Cartan subalgebra $\mathfrak h ^\ast$, so the fixed points on $\cal O$ are the intersection $\cal O \cap \mathfrak h ^\ast$. The set of elements of $G$ which preserve $\cal O \cap \mathfrak h ^\ast$ is precisely $N(T)$. Generically, the stabilizer of the $G$ action at these points is the maximal torus $T$ corresponding to $\mathfrak{h}$. Thus, the points of $\cal O \cap \mathfrak h ^\ast$ are in bijection with $N(T)/T$. This is the Weyl group of $G$. In general, The fixed points of $T$ are orbits of the Weyl group. The result follows.
The structural theory of Lie groups has two central observations. First, lie groups are essentially equivalent to their lie algebras. Second, all the information of a group $G$ is contained in its Weyl group $W$ acting on the Cartan subalgebra $\mathfrak{h}$. For example, the classification of Lie groups proceeds by classifying possible Weyl groups. All this to say, reducing the coadjoint orbits to convex hulls of the Weyl group is a classic strategy.
Now we do morse theory. We choose a function $f_X : \cal P \to \mathbb{R}$, $f_X(m) = \mu(m)(X)$ for $X \in \mathfrak{h}$. The Hamiltonain vector field for $f_X$ generates a flow on $\cal O$ corresponding to the constant vector field $X$ on the maximal torus. $f_X$ is computed by first projecting $\cal O$ to its moment polytope, then orthogonally projecting to the subspace spanned by $X^\flat$. We choose $X$ generically, so that its flow is dense in $T$. Then, $f_X$ has fixed points exactly where $T$ does, and all the information of the torus action is encoded in $f_X$. On the polytope level, this projects to a subspace such that all faces of the toric polytope do not project to the same value. In this case, this happens if $X$ is not preserved by any element of the Weyl group.
The index of each of these critical points counts how many of the edges from each corner are pointing towards $X$, minus the number pointing away from $X$, times two. In particular, it is always even. This means $f_X$ is a perfect morse function, so we can instantly compute the homology of $\cal O$ as a vector space. The closure of the stable manifolds of each critical point of $f_X$ generate the homology, with the homology grading equaling the index of the critical point. In terms of polytopes, this consists of the subpolytope spanned by all cells of the original polytope, adjacent to the fixed point, in the positive $X$ direction. (this morse-theoretic approach is behind Atiyah’s proof of the convexity of the moment map image). This is called the Burhat decomposition of flag manifolds, and it reduces to the Schubert cells of Grassmanians.
This gleans us a lot of information on coadjoint orbits. Fortunately, coadjoint orbits include all symmetric spaces $G_\mathbb{C}/P_{\mathbb{C}}$ for $P$ a parabolic subgroup (Where $P$ is parabolic iff $G_\mathbb{C}/P_{\mathbb{C}}$ is an algebraic variety). This includes the largest symmetric space, the flag variety $G_\mathbb{C}/B_{\mathbb{C}} = G/T$, which for $G=U(n)$ is the full flag variety. It also includes partial flag varieties, like Grassmanians.
The results above I think are fairly standard, but I learned them following these notes. They take a lower brow approach than me, developing the needed Lie theory along the way. Above, I repackaged these statements using toric geometry.
Analogues for loop groups
The lie group $LG$ of a simple group $G$ behaves in many respects like a simple lie group with rank one greater than that of $G$. This manifests directly via a group action of $G$ on $LG$, and of $S^1$ on $LG$. Let $\gamma:S^1 \to G$ be a loop in $G$, corresponding to a point in $LG$.
- The “momentum operator”, which measures the average direction of a loop:
- $p:LG \to \mathfrak{g} \quad p(\gamma) = \int_{S^1} \gamma’$
- The “energy operator”, which measures the average velocity of a loop
- $E:LG \to \mathbb{R} \quad E(\gamma) = \int_{S^1} \vert \gamma’ \vert^2$
When dualized, $p$ is the momentum map for the left multiplication action of $G$ on $LG$. Likewise, we can identify $\mathbb{R}$ with the dual of the lie algebra of $S^1$, and $E$ is a function whose hamiltonian vector field generates rigid rotation of the loop in the $S^1$ direction.
The maximal torus in $LG$ is generated by rigid multiplications by $T\subset G$, and rigid rotations in the $S^1$ direction, which we denote by $\mathbb{T}$. In total, it is $T\times \mathbb{T}$. Projecting $p$ onto the the cartan algebra $\mathfrak{h}$, we can look at the image of coadjoint orbits of $LG$. The resulting shape is convex, but not bounded. It is described in Convexity and Loop groups by Atiyah and Pressley.
Once again, the fixed points of the $T \times \mathbb{T}$ action on the coadjoint orbits are the intersections of the orbit with $\mathfrak{h}\times \mathbb{R}$, which we identify with orbits of $N(T\times \mathbb{T})/T\times \mathbb{T}$: The affine Weyl group $W_{aff}$. This consists of two parts
- parts inheret to $G$, which gives the weyl group $W$
- The parts combining $\mathfrak