TL;DR
This paper investigates the properties of wall-crossing functors in the representation theory of algebraic groups over fields of positive characteristic, focusing on their derived functors and their effects on global sections.
Contribution
It introduces a conjecture about higher derived functors of wall-crossing and proves acyclicity of certain modules in this setting.
Findings
The kernel of the wall-crossing transformation is a left exact functor.
Higher derived functors beyond the first are conjectured to vanish on global sections.
Principal indecomposable modules for Frobenius subgroups are shown to be acyclic.
Abstract
The setting is the representation theory of a simply connected, semisimple algebraic group over a field of positive characteristic. There is a natural transformation from the wall-crossing functor to the identity functor. The kernel of this transformation is a left exact functor. This functor and its first derived functor are evaluated on the global sections of a line bundle on the flag variety. It is conjectured that the derived functors of order greater than one annihilate the global sections. Also, the principal indecomposable modules for the Frobenius subgroups are shown to be acyclic.
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