  Let $\pi:\widetilde{\mathcal N}\to\mathcal N$ be the Springer resolution of
the nilpotent cone for a semisimple connected algebraic group $G$ over $\mathbb
C$ and $k$ be an arbitrary field. What happens to $\pi_*k[\dim\mathcal N]$ if
the decomposition theorem fails for it? We show that in this case, some
additional (with respect to the case ${\rm char}\;k=0$) composition factors of
this direct image in the (abelian) category of perverse sheaves may emerge.
These factors emerge from the $Z_G(x)/Z_G(x)^0$-composition factors of the
radicals of certain intersection forms and from that of the top comohologies of
Springer fibres (in the non-semisimple case).
