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Let $X$ be a rational curve and $E$ a stable parabolic vector bundle on $X$. Is the sheaf of parabolic endomorphisms (i.e the endomorphisms preserving the flag) of the sheaf $E$ also stable (or atleast semi-stable)?

Any suggestions or references will be helpful.

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    $\begingroup$ As your title indicates, you can only hope for semi-stability, since the endomorphisms sheaf has a trivial direct summand. $\endgroup$
    – abx
    Commented Apr 19, 2018 at 19:05
  • $\begingroup$ @abx but is it semi-stable atleast? Can you give a reference. $\endgroup$
    – user43198
    Commented Apr 19, 2018 at 19:56

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