  Let $G$ be a reductive complex algebraic group, $T$ a maximal torus of $G$,
$B$ a Borel subgroup of $G$ containing $T$, $\Phi$ the root system of $G$
w.r.t. $T$, $W$ the Weyl group of $\Phi$. Denote by $\Fo = G/B$ the flag
variety, by $X_w$ the Schubert subvariety of $\Fo$ associated with an element
$w\in W$, and by $C_w$ the tangent cone to $X_w$ at the point $p = eB$. Then
$C_w$ is a subscheme of the tangent space $T_pX_w\subseteq T_p\Fo$. Suppose
$w$, $w'$ are distinct involutions in $W$. Using the so-called Kostant--Kumar
polynomials, we show that if every irreducible component of $\Phi$ is of type
$A_n$, $F_4$ or $G_2$, then $C_w$ and $C_{w'}$ do not coincide.
