  We determine the graded composition multiplicity in the symmetric algebra
S(V) of the natural GL_n(q)-module V, or equivalently in the coinvariant
algebra of V, for a large class of irreducible modules around the Steinberg
module. This was built on a computation, via connections to algebraic groups,
of the Steinberg module multiplicity in a tensor product of S(V) with other
tensor spaces of fundamental weight modules.
