  We give a new interpretation of O'Grady's filtration on the $CH_0$ group of a
$K3$ surface. In particular, we get a new characterization of the canonical
0-cycles $kc_X$ : this is the only 0-cycle on $X$ whose orbit under rational
equivalence is of dimension $k$. Using this, we extend results of Huybrechts
and O'Grady concerning Chern classes of simple vector bundles on $K3$ surfaces.
