⇤Math diary
Jun 12, 2023:
I realized that to understand the BGG resolution, I need to understand the cell structure of the flag manifold for loop groups. In the finite dimensional case, I can understand this well using symplectic geometry, moment polytopes, and morse theory. I describe this in entry 15. Essentially, the flag manifold is itself a coadjoint orbit, so it naturally sits inside the lie algebra. Projecting to a cartan algebra gives the hamiltonian functions generating the action of the maximal torus, and the image is a convex polytope. The vertices of the polytope are an orbit of the weyl group. Using morse theory, this gives a cell decomposition with cells indexed by the Weyl group, the burhat decomposition.
In infinite dimensions, this story must be modified, as in Convexity and Loop groups by Atiyah and Pressley. The based loop group $\Omega G = LG / G$ is a coadjoint orbit in the lie algebra of $LG$. Formally applying atiyah convexity, we see the image polytope in the cartan subalgebra (plus $\mathbb{R}$ ) is a unbounded polytope. Moreover, the vertices of this polytope correspond to elements of the affine Weyl algebra, and these index a cellular decomposition of $\Omega G$. I believe this decomposition is what we need for the BGG resolution on loop space. Unfortunately, this is for the based loop space, not the full flag manifold. These are very similar critters, but a little bit different, so I don’t know f they have the same cell structure. I need to go through the Atiyah-Pressley paper, but I think it’s the key for me really understanding semi-infinite flag manifolds.