Trivial Witt groups of flag varieties

Journal of pure and applied algebra, 216, issue 2, 2012, p. 404-406
(1^st online appearance of the preprint on the 22-3-2011)

Let G be a split semi-simple linear algebraic group over a field, let P be a parabolic subgroup and let L be a line bundle on the projective homogeneous variety G/B. We give a simple condition on the class of L in Pic(G/P)/2 in terms of Dynkin diagrams implying that the Witt groups Wⁱ(G/P,L) are zero for all integers i. In particular, if B is a Borel subgroup, then Wⁱ(G/B,L) is zero unless L is trivial in Pic(G/B)/2.