LSL2 Polytopes (by walls) are parabolas

My previous treatment of LSL2 polytopes only took into account the real roots.  Given a collection of integers associated to each real chamber weight we defined as the intersection of half-spaces facing in the direction of and displaced by .  This was some three dimensional bi-infinite shape, not similar to weights of irreps, which should be restricted to a single central character.

If you add an imaginary chamber weight though, and instead of imposing an inequality, require equality then we will get parabolic polytopes.  Furthermore each edge has integer slope which is an observed phenomenon.

Imposing equality rather than inequality may be loosely justified a few ways.  First, in some sense there are two imaginary chamber weights in opposite directions — but they are fundamentally the same so should have corresponding numbers. Then the inequalities pointing in opposite directions yield equality.  Second, given the two dimensional polytope (using one equality) and taking some non-degeneracy assumptions we can always reconstruct the three dimensional polytope (defined only by inequalities).

Unfortunately none of the Plucker relations listed in Kamnitzer apply in this situation.  And there’s still the problem of not having a “longest element” of the Weyl group.