  In the present paper, we discuss applications of the derived completion
theorems proven in our previous two papers. One of the main applications is to
Riemann-Roch problems for forms of higher equivariant K-theory, which we are
able to establish in great generality both for equivariant G-theory and
equivariant homotopy K-theory with respect to actions of linear algebraic
groups on normal quasi-projective schemes over a given field. We show such
Riemann-Roch theorems apply to all toric and spherical varieties.
  We also obtain Lefschetz-Riemann-Roch theorems involving the fixed point
schemes with respect to actions of diagonalizable group schemes. We also show
the existence of certain spectral sequences that compute the homotopy groups of
the derived completions of equivariant G-theory starting with equivariant
Borel-Moore motivic cohomology.
