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Edge α − β and (n 1 , n 2 ) = (even, even). L = {(ϕ, ϕ, ϕ)}. 17) x02 (x12 + x32 ), x02 (x12 + x22 ), x02 (sin(ϕ) x0 x2 − i x1 x3 ), x02 (sin(ϕ) x0 x1 + i x2 x3 ), x03 x2 , x03 x1 , x03 x3 }. Thus the characteristic variety contains the hyperplane x0 = 0. For x0 = 0 the last three minors show that all other coordinates vanish and this gives an additional point, not in the above hyperplane. The correspondence σ is the identity. 9. Edge α ⊥ β and (n 1 , n 2 ) = (even, even). C+ = {(ϕ, ϕ, 0)}. In that case the list of minors simplifies (with non-zero scale factors removed) to the following: { x0 (x12 + x22 ) x3 , (x12 + x22 ) (−i cos(ϕ) x0 x2 + sin(ϕ) x1 x3 ), (sin(ϕ) x0 x2 + i cos(ϕ) x1 x3 ) (x02 + x32 ), (x02 x22 − x12 x32 ), x1 x2 (x02 + x32 ), (x12 + x22 ) (cos(ϕ) x0 x1 − i sin(ϕ) x2 x3 ), (sin(ϕ) x0 x1 − i cos(ϕ) x2 x3 ) (x02 + x32 ), x1 x2 (x02 + x32 ), (x02 x12 − x22 x32 ), sin(4 ϕ) (x12 + x22 ) (x02 + x32 ), + x22 ) (i sin(ϕ) x0 x2 + cos(ϕ) x1 x3 ), + x22 ) (sin(ϕ) x0 x1 + i cos(ϕ) x2 x3 ), (cos(ϕ) x0 x2 − i sin(ϕ) x1 x3 ) (x02 + x32 ), cos(ϕ) x0 x1 + sin(ϕ) x2 x3 ) (x02 + x32 ), x0 (x12 + x22 ) x3 }.

And (n 1 , n 2 ) = (even, even) gives the line C+ of Case 7 below. • α ⊥ β and (n 1 , n 2 ) = (odd, odd) gives the line C− of Case 8 below. 3 that δ(ϕ) = 0. Then if ϕ ∈ H(α,n) and n is even (resp. odd) the root α is one of the differences ϕk − ϕl (resp. ϕk ). Thus up to permutations of the ϕk one obtains one of the Cases 1)-6). The complete table giving the geometric datas is the following: 4 We are grateful to Marc Bellon for pointing out the subtelty of case 9) which was incomplete in an earlier version.

1 , ϕ2 , ϕ3 , ϕ1 − ϕ2 , ϕ2 − ϕ3 , ϕ3 − ϕ1 }. Moreover the periodicity lattice of ϕ is (π Z)3 which is specified by {ϕ, α(ϕ) ∈ π Z, ∀ α ∈ }= ϕ. 34) We now want to relate more precisely the above situation with canonical objects (root systems, alcoves, chambers, affine Weyl group, nodal vectors . ) associated to the following data (G, T): G = P S O(6), T = Maximal torus T D / ± 1. 35) We use the natural parametrization of the Lie algebra Lie (T), (ξ ) = ξ1 β12 + ξ2 β34 + ξ3 β56 . 1 3), the parameter ψ that appears in the transition ϕ → ψ of Eq.

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Communications In Mathematical Physics - Volume 281 by M. Aizenman (Chief Editor)

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