I hope that this is the appropriate place for asking about a step I don't understand in a proof which I think is due to a lack of knowledge. This is a step in Drinfeld-Simpson's paper: ``*$B$ structures on $G$-bundles and local triviality*" in which they showed that under some nice conditions, every $G$-bundle on a curve admits a $B$-reduction. Let me get to the specific statement.


Suppose that $X$ is a smooth projective curve over an algebraically closed field $k$. Let $\mathcal{P}$ be a $G$-bundle on $X$ where $G$ is a linear algebraic group over $k$.


Now, some notation: let $B$ be a Borel of $G$, $\alpha\_i: H \rightarrow G\_m$ a simple root. If $\mathcal{E}$ is a $B$-bundle then I can produce the $G\_m$-bundle associated to $\mathcal{E}$ and call its degree $\deg\_i(\mathcal{E})$ just as a line bundle over a smooth projective curve.


The problem set up: suppose I now have another $G$-bundle $\mathcal{P}'$ and (1) I have a trivialization of both $\mathcal{P}$ and $\mathcal{P}'$ on an open subscheme of $X$ and I have chosen $h$ an isomorphism between them and (2) a $B$-reduction on $\mathcal{P}$ (with corresponding $B$-bundle $\mathcal{E}$) which induces a $B$-reduction on $\mathcal{P}'$ with corresponding $B$-bundle $\mathcal{E}'$.


**Question**:


Why does there exist a number $c$ such that $-c < \deg\_i(\mathcal{E}) - \deg\_i(\mathcal{E}') < c$ depending on the singularities of $h$? (as claimed)


I suppose I don't see how these numerical data relate to one another.



A step in the proof of the Drinfeld-Simpson theorem
Fix two principal $G$-bundles $P\_G$ and $P'\_G$ on $X$ along with an identification $\beta$ away from a divisor $D$. Also fix a simple root $\alpha\_i$.


Let $P\_B$ and $P\_B'$ be arbitrary $B$-reductions (of $P\_G$ and $P\_G'$ respectively) compatible with $\beta$. We write
$$
 L = P\_B \overset{B}{\times} \alpha\_i \qquad E = P\_G \overset{G}{\times} V\_{\alpha\_i},
 $$
and similarly for $L'$ and $E'$. So $P\_B$ exhibits $L$ (resp. $L'$) as a "highest weight" line subbundle of $E$ (resp. $E'$).


By use of $\beta$, we may write
$$
 E(-m \cdot D) \subset E' \subset E(n \cdot D)
 $$
for some integers $m$ and $n$ independent of $P\_B$ and $P'\_B$. It follows that
$$
 L(-m \cdot D) \subset L' \subset L(n \cdot D).
 $$
Taking degrees gives a bound of the desired type.


