Kostant–Kumar polynomials and tangent cones to Schubert varieties for involutions in $A_n$, $F_4$ and $G_2$

Let $G$ be a complex reductive algebraic group and $W$ its Weyl group. We prove that if $W$ are of type $A_n$, $F_4$ or $G_2$ and $w,w'$ are disjoint involutions in $W$, then the corresponding Kostant–Kumar polynomials do not coincide. As a consequence, we deduce that the tangent cones to the Schubert subvarieties $X_w$, $X_{w'}$ of the flag variety of $G$ do not coincide, too.