  For a smooth quasi-projective scheme $X$ over a field $k$ with an action of a
reductive group, we establish a spectral sequence connecting the equivariant
and the ordinary higher Chow groups of $X$. For $X$ smooth and projective, we
show that this spectral sequence degenerates, leading to an explicit relation
between the equivariant and the ordinary higher Chow groups. We obtain several
applications to algebraic $K$-theory.
  We show that for a reductive group $G$ acting on a smooth projective scheme
$X$, the forgetful map $K^G_i(X) \to K_i(X)$ induces an isomorphism
$K^G_i(X)/{I_G K^G_i(X)} \cong K_i(X)$ with rational coefficients. This
generalizes a result of Graham to higher $K$-theory of such schemes. We prove
an equivariant Riemann-Roch theorem, leading to a generalization of a result of
Edidin and Graham to higher $K$-theory. Similar techniques are used to prove
the equivariant Quillen-Lichtenbaum conjecture.
