  We develop the affine sieve in the context of orbits of congruence subgroups
of semi-simple groups acting linearly on affine space. In particular we give
effective bounds for the saturation numbers for points on such orbits at which
the value of a given polynomial has few prime factors. In many cases these
bounds are of the same quality as what is known in the classical case of a
polynomial in one variable and the orbit is the integers. When the orbit is the
set of integral matrices of a fixed determinant we obtain a sharp result for
the saturation number, and thus establish the Zariski density of matrices all
of whose entries are prime numbers. Among the key tools used are explicit
approximations to the generalized Ramanujan conjectures for such groups and
sharp and uniform counting of points on such orbits when ordered by various
norms.
