  Kashiwara conjectured that the hard Lefshetz theorem and the semisimplicity
theorem hold for any semisimple perverse sheaf M on a variety over a field of
characteristic 0. He also conjectured that if you apply to such M the nearby
cycle functor corresponding to some function then the successive quotients of
the monodromy filtration are semisimple. We prove that these conjectures would
follow from de Jong's conjecture on representations modulo l of the arithmetic
fundamental group of a variety over a finite field.
