  Let G be a reductive algebraic group over a field k. When k=C, R.K.Brylinski
constructed a filtration of weight spaces of a G module, using the action of a
principal nilpotent element of the Lie algebra, and proved that this filtration
corresponds to Lusztig's q-analogue of the weight multiplicity. Later, Ginzburg
discovered that this filtration has an interesting geometric interpretation via
the geometric Satake correspondence. The goal of this article is to generalize
these results to positive characteristics.
