  We consider a certain class of Schubert varieties of the affine Grassmannian
of type A. By embedding a Schubert variety into a finite-dimensional
Grassmannian, we construct an explicit basis of sections of the basic line
bundle by restricting certain Pl\"ucker co-ordinates.
  As a consequence, we write an explicit set of generators for the degree-one
part of the ideal of the finite-dimensional embedding. This in turn gives a set
of generators for the degree-one part of the ideal defining the affine
Grassmannian inside the infinite Grassmannian which we conjecture to be a
complete set of ideal generators.
  We apply our results to the orbit closures of nilpotent matrices. We describe
(in a characteristic-free way) a filtration for the coordinate ring of a
nilpotent orbit closure and state a conjecture on the SL(n)-module structures
of the constituents of this filtration.
