On the decomposition numbers of SO8+(2f)

Let q=2f, and let G=SO8+(q) and U be a Sylow 2-subgroup of G. We first describe the fusion of the conjugacy classes of U in G. We then use this information to prove the unitriangularity of the ℓ-decomposition matrices of G for all ℓ≠2 by inducing certain irreducible characters of U to G; the characters of U of degree q3/2 play here a major role. We then determine the ℓ-decomposition matrix of G in the case ℓ|q+1, when ℓ≥5 and (q+1)ℓ>5, up to two non-negative indeterminates in one column.