Composition factors for the Springer resolution

Let $π:\widetilde{\mathcal N}\to\mathcal N$ 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 $π_*k[\dim\mathcal N]$ 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).

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