We prove some analogues of Schur's lemma for endomorphisms of extensions in Tannakian categories. More precisely, let $\mathbf{ T}$ be a neutral Tannakian category over a field of characteristic zero. Let $E$ be an extension of $A$ by $B$ in $\mathbf{ T}$. We consider conditions under which every endomorphism of $E$ that stabilizes $B$ induces a scalar map on $A\oplus B$. We give a result in this direction in the general setting of arbitrary $\mathbf{ T}$ and $E$, and then a stronger result when $\mathbf{ T}$ is filtered and the associated graded objects to $A$ and $B$ satisfy some conditions. We also discuss the sharpness of the results.
Categories: math.AG math.NT math.RT