On tangent cones to Schubert varieties in type $D_n$

Abstract

Let $G$ be a complex reductive algebraic group, $T$ a maximal torus in $G$, $B$ a Borel subgroup of $G$ containing $T$, $W$ the Weyl group of $G$ with respect to $T$. Let $w$ be an element of $W$. Denote by $X_w$ the Schubert subvariety of the flag variety $G/B$ corresponding to $w$. Let $C_w$ be the tangent cone to $X_w$ at the point $p=eB$ (we consider $C_w$ as a subscheme of the tangent space to $G/B$ at $p$). In 2011, D.Yu. Eliseev and A.N. Panov computed all tangent cones for $G=SL(n)$, $n<6$. Using their computations, A.N. Panov formulated the following Conjecture: if $w$, $w'$ are distinct involutions in $W$, then $C_w$ and $C_{w'}$ do not coincide. In 2013, D.Yu. Eliseev and the first author proved this conjecture in types $A_n$, $F_4$ and $G_2$. Later M.A. Bochkarev and the authors proved this conjecture in types $B_n$ and $C_n$. In this paper we prove the conjecture in type $D_n$ in the case when $w$, $w'$ are basic involutions.