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Let G be a reductive algebraic group over a field k. When k=C, R.K.Brylinskiconstructed a filtration of weight spaces of a G module, using the action of aprincipal nilpotent element of the Lie algebra, and proved that this filtrationcorresponds to Lusztig's q-analogue of the weight multiplicity. Later, Ginzburgdiscovered that this filtration has an interesting geometric interpretation viathe geometric Satake correspondence. The goal of this article is to generalizethese results to positive characteristics.

Filtrations of tilting modules and costalks of parity sheaves