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

Let $G$ be a reductive complex algebraic group, $T$ a maximal torus of $G$, $B$ a Borel subgroup of $G$ containing $T$, $Φ$ the root system of $G$ w.r.t. $T$, $W$ the Weyl group of $Φ$. 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 $Φ$ is of type $A_n$, $F_4$ or $G_2$, then $C_w$ and $C_{w'}$ do not coincide.

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