Stability of representation cohomology sheaves on moduli spaces of bundles

We prove that the tangent sheaf of ${\text{SU}}_C(n,L)$ is slope stable for every smooth complex projective curve $C$ of genus $g\ge 2$, every $n\ge 2$, and every determinant $L$. For $g\ge 3$, we prove a more general stability theorem on moduli spaces of principal bundles for connected reductive groups, using cohomology sheaves associated with representations in place of tangent sheaves. The adjoint representation gives tangent sheaf stability for the case of almost simple groups. The moduli spaces may be singular, and stability holds with respect to every ample Cartier polarization.