Facets of the fundamental parallelepiped carrying a unitary spherical representation, and their closure relations. Unitarity is constant on each facet. Rank 1 and 2 can be drawn outright; the rest are read off the poset.
The FPP itself. Every point is an infinitesimal character γ, and the region is the parallelepiped spanned by the fundamental weights; the affine walls ⟨γ,α∨⟩ ∈ ℤ cut it into facets. Shaded facets are the ones whose spherical representation is unitary — click one to inspect it.
The closure poset collapsed by the Cartan type of the integral root system Φ(γ), which determines the dimension. Click a type to trace what it degenerates to and what degenerates to it.
Every component is collapsible, so each is contractible and the whole set is homotopy equivalent to its set of components. Drawn here as 1-skeleta: dots are 0-dimensional facets, segments 1-dimensional ones. Colour is the vertex of the fundamental alcove the point is Waff-conjugate to.