  It is known that the cotangent bundle $\Omega_Y$ of an irreducible Hermitian
symmetric space $Y$ of compact type is stable. Except for a few obvious
exceptions, we show that if $X \subset Y$ is a complete intersection such that
$Pic(Y) \to Pic(X)$ is surjective, then the restriction $\Omega_{Y|X}$ is
stable. We then address some cases where the Picard group increases by
restriction.
