KO-theory of complex flag varieties of ordinary type

We compute the topological Witt groups of every complex flag manifold of ordinary type, and thus the interesting (i.e. torsion) part of the KO-groups of these manifolds. Equivalently, we compute Balmer's Witt groups of each flag variety of ordinary type over an algebraically closed field of characteristic not two. Our computation is based on an approach developed by Zibrowius. For types A, B and C, we obtain a full description not only of the additive but also of the multiplicative structure of the graded Witt rings.