The orbits of B on G/B, “Bruhat cells”, are simple as varieties (they’re just vector spaces) but have a very interesting and comprehensible stratifications by singular subvarieties, the “Kazhdan-Lusztig varieties”. I’ll define a “Bruhat atlas” on a stratified manifold as an atlas consisting of Bruhat cells, such that the chart maps correspond the stratifications. In particular, the poset is identified with an order ideal in a Bruhat order, which is already very restrictive.

The motivating example was the positroid stratification of the Grassmannian, whose strata we indexed by affine permutations in [Knutson-Lam-Speyer]; this was souped up to geometry in Snider’s 2010 thesis. I’ll explain how to put Bruhat atlases on general G/P (and more generally, Q-orbits on G/P) and on wonderful compactifications of groups. This work is joint with Jiang-Hua Lu and Xuhua He.

For inductive classification purposes, one wants to study not just the manifolds but their strata, which inherit “Kazhdan-Lusztig atlases”. The first nontrivial problem is to classify equivariant K-L atlases on smooth projective toric surfaces. These have been largely classified by Bal\’azs Elek in his thesis: there are 18 or 19 simply-laced atlases, and < 8000 non-simply-laced.