In Lusztig's papers from 1985-1986 that invented the theory of character sheaves, he proved (in nearly all cases) a remarkable property of cuspidal perverse Q-sheaves on the nilpotent variety: they are 'clean', meaning that their stalks vanish outside a single orbit. This property is crucial to making character sheaves computable by an algorithm, and it is a precursor of various 'block decompositions' of the derived category studied by various authors (Gunningham, Rider, Russell, and others) later. About 10 years ago, Mautner conjectured that these perverse sheaves remain clean after reduction modulo p (with some exceptions for small p). In this talk, I will discuss the history and context of the cleanness phenomenon, along with recent progress on Mautner’s conjecture.
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