springer_table_G.at

for |rd="G2".simply_connected|, the command |orbits(rd)| returns 5 orbits,
whose diagrams are, in order, [0,0],[0,1],[1,0],[0,2],[2,2]. The springer map
send these respectively to the (full) sign, short-root-reflection sign,
triangluar reflection, reflection, and trivial representations of W(G2) (in
the numbering of irreducibles fixed in |character_table_G|, numbers 3,1,4,5,0
respectively). The dual map sends the same orbits to just 3 dual orbits, whose
stratified diagrams (so the dual Dynkin diagram order is reversed) are
respectively [2,2],[0,2],[0,2],[0,2],[0,0].

Below we take an ad hoc approach too the dual map, testing explicitly for the
two extreme cases. Since we are supposed to ignore any radical component of
|O.H|, we avoid testing |O.H| directly for being zero or |rd.two_rho_check|,
computing the diagram instead which will do the required ignoring.