  Let $G$ be a simple simply-laced algebraic group of adjoint type over the
field $\mathbb{C}$ of complex numbers, $B$ be a Borel subgroup of $G$
containing a maximal torus $T$ of $G.$ In this article, we show that
$\omega_\alpha$ is a minuscule fundamental weight if and only if for any
parabolic subgroup $Q$ containing $B$ properly, there is no Schubert variety
$X_{Q}(w)$ in $G/Q$ such that the minimal parabolic subgroup $P_{\alpha}$ of
$G$ is the connected component, containing the identity automorphism of the
group of all algebraic automorphisms of $X_{Q}(w).$
