  Let X be a smooth separated geometrically connected variety over F_q and
f:Y-> X a smooth projective morphism. We compare the invariant dimensions of
the l-adic representation V_l and the F_l-representation \bar V_l of the
geometric \'etale fundamental group of X arising from the sheaves R^wf_*Q_l and
R^wf_*Z/lZ respectively. These invariant dimension data is used to deduce a
maximality result of the geometric monodromy action on V_l whenever \bar V_l is
semisimple and l is sufficiently large. We also provide examples for \bar V_l
to be semisimple for l>>0.
