  The paper was motivated by a question of Vilonen, and the main results have
been used by Mirkovic and Vilonen to give a geometric interpretation of the
dual group (as a Chevalley group over Z) of a reductive group. We define a
quasi-reducitve group over a discrete valuation ring R to be an affine flat
group scheme over R such that (i) the fibers are of finite type and of the same
dimension; (ii) the generic fiber is smooth and connected, and (iii) the
netural component of the reduced special fiber is a reductive group.
  We show that such a group scheme is of finite type over R, the generic fiber
is a reductive group, the special fiber is connected, and the group scheme is
smooth over R in most cases, for example when the residue characteristic is not
2, or when the generic fiber and reduced special fiber are of the same type as
reductive groups. We also obtain results about group schemes over a Dedekind
scheme or a noetherian scheme. We show that in residue characteristic 2 there
are indeed non-smooth quasi-reductive groups and they can be classified when R
is strictly henselian.
