slope of the bundles.

For any vector bundle $E$, there exists a canonical filtration of $E$ by $\OO_X$-coherent subsheaves known as its Harder-Narasimhan filtration (denoted HN$(E)$). $$0 \subset E_1 \subset \cdots \subset E_l = E$$ with the property that successive quotients $E_i/E_{i-1}$ are $\mu$-semistable and $\mu (E_i/E_{i-1}) > \mu
(E_{i+1}/E_i)$ for all $1 \leq i \leq l$.

Define $\mu_{max}(E) = \mu (E_1)$ and by $\mu_{min}(E)=\mu(E_l/E_{l-1})$

The quantity $\mu_{max}(E)-\mu_{min}(E)$ known as the [**instability degree**]{} is a measure of the instability of the vector bundle.

A principal $G$-bundle $E$ over $X$ is said to be semistable if for any reduction of structure group to a parabolic $P$ of $G$ and any dominant character on $P$, the induced line bundle has degree $\leq$ 0.\
$~$ Equivalently, a principal bundle $E$ on $X$ is said to be semistable is for any reduction of structure group to a parabolic $P$ of $G$, the pullback of the relative tangent bundle of $E(G/P)$ over $X$, via the section $\sigma: X \rightarrow E(G/P)$ corresponding to this reduction is a vector bundle on $X$ of degree $\geq$ 0.

\[H-N filtration\] Let $E$ be a principal $G$ bundle on $X$. Let $E_P$ be a reduction of structure group af $E$ 