The GLn(q)-module structure of the symmetric algebra around the Steinberg module

We determine the graded composition multiplicity in the symmetric algebra S•(V) of the natural GLn(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.