Tangent cones to Schubert varieties in types $A_n$, $B_n$ and $C_n$

Abstract

Let $G$ be a complex reductive group, $T$ be a maximal torus of $G$, $B$ be a Borel subgroup of $G$ containing $T$, $W$ be the Weyl group of $G$ with respect to $T$. To each element $w$ of $W$ one can associate the Schubert subvariety $X_w$ of the flag variety $G/B$, the tangent cone to $X_w$ at the identity point $p$ considered as a subcheme of the tangent space $T_p(G/B)$, and the reduced tangent cone to $X_w$ at $p$ considered as a subvariety of $T_p(G/B)$. Let $w_1$, $w_2$ be distinct involutions in $W$. We prove that if $G$ is of type $B_n$ or $C_n$, then the tangent cones corresponding to $w_1$ and $w_2$ are distinct. We also prove that if $G$ is of type $A_n$ or $C_n$, then the reduced tangent cones corresponding to $w_1$ and $w_2$ are distinct.