  When a reductive group $G$ acts linearly on a complex projective scheme $X$
there is a stratification of $X$ into $G$-invariant locally closed subschemes,
with an open stratum $X^{ss}$ formed by the semistable points in the sense of
Mumford's geometric invariant theory which has a categorical quotient $X^{ss}
\to X//G$. In this article we describe a method for constructing quotients of
the unstable strata. As an application, we construct moduli spaces of sheaves
of fixed Harder-Narasimhan type with some extra data (an '$n$-rigidification')
on a projective base.
