Let
X
X
be a smooth, irreducible complex projective curve of genus
g
≥
2
g \geq 2
and
U
μ
1
(
n
,
d
)
U_{{\mu _1}}(n,d)
be the moduli scheme of indecomposable vector bundles over
X
X
with fixed Harder-Narasimhan type
σ
=
(
μ
1
,
μ
2
)
\sigma =(\mu _1, \mu _2)
. In this paper, we give necessary and sufficient conditions for a vector bundle
E
∈
U
μ
1
(
n
,
d
)
E\in U{{\mu _1}}(n,d)
to have
C
[
x
1
,
…
,
x
k
]
/
(
x
1
,
…
,
x
k
)
2
\mathbb {C}[x_1,\dots , x_k]/(x_1,\dots , x_k)^2
as its algebra of endomorphisms. By fixing the dimension of the algebra of endomorphisms, we obtain a stratification of
U
μ
1
(
n
,
d
)
U {\mu _1} (n, d)
, where each stratum
U
μ
1
(
n
,
d
,
k
)
U {\mu _1} (n, d, k)
is an algebraic variety, moreover, it is a coarse moduli space. A particular case of interest arises when the unstable bundles are simple. In such a case, the moduli space is fine. The topological properties of
U
μ
1
(
n
,
d
,
k
)
U_{\mu _1} (n, d, k)
will depend on the generality of the curve
X
X
. These results differ from the corresponding results for the moduli space of stable bundles, where non-emptiness, irreducibility, dimension are independent of the curve.