Composition factors for the Springer resolution

TL;DR

This paper investigates how the decomposition theorem's failure affects the composition factors of the Springer resolution's direct image, revealing additional factors linked to intersection forms and Springer fibres in positive characteristic.

Contribution

It demonstrates the emergence of extra composition factors in the direct image of the Springer resolution when the decomposition theorem does not hold, especially in positive characteristic.

Findings

Additional composition factors appear in the direct image in positive characteristic.

These factors relate to the radicals of intersection forms and Springer fibre top homologies.

Failure of the decomposition theorem leads to more complex perverse sheaf structures.

Abstract

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).