  Let $G$ be a compact Lie group and $T$ its maximal torus. In this paper, we
try to compute $gr_{\gamma}^*(G/T)$ the graded ring associated with the gamma
filtration of the complex $K$-theory $K^0(G/T)$ for $G=Spin(n)$. In particular,
we give a counterexample for a conjecture by Karpenko when $G=Spin(17)$. The
arguments for $E_7$ in $\S 11$ of the old version were not correct, and they
are deleted in this version.
