on $E(G/P)_\circ$ with respect to the polarization $E(\mathcal L)_\circ$. Then $T_\sigma$ has degree $\geq 0$.\
$~$ In other words this reduction of structure group does not contradict semistability of $E(Gl(V)$.

[\[HC\]]{} (See \[RR\]) Let $E$ be a semistable $G$-bundle. Suppose for every reduction to a parabolic $P$ in $Gl(V)$, the instability parabolic associated to this reduction is rational (defined over $k(X)$), then the induced $Gl(V)$ bundle is semistable.

[\[field of definition\]]{} (See \[HC\], Proposition 4.5) Let $G$ be a reductive algebraic group defined over an arbitrary field $K$ (not neccessary algebraically closed) acting on a projective variety $M$ defined over $K$. Then there exists an integer $t$ such that given any $K$-valued point $m \in M$ which is not semistable its instability parabolic $P(m)$ is defined over $K^{p^{-t}}$.

[**The method of Holla-Coiai**]{}

In this section we briefly explain the method of Holla-Coiai for proving the existence of the integer $t$ in proposition \[field of definition\]. We will be brief and sketchy in this exposition. Let $G$ and $M$ be as in above proposition \[field of definition\]. Let $\mathcal L$ be a linearized very ample line bundle on $M$ giving a $G$-equivariant embedding $i: M \hookrightarrow {\mathbb {P}}(H^\circ(M,{\mathcal L}))={\mathbb
{P}}(V)$\
For 