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 \(T\), \(\Phi\) 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.