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

TL;DR

This paper proves that for certain types of root systems, the tangent cones to Schubert varieties at involutions are uniquely determined by the involution, using Kostant--Kumar polynomials.

Contribution

It establishes the distinctness of tangent cones at involutions in specific root system types using Kostant--Kumar polynomials.

Findings

Tangent cones at involutions are unique for types A_n, F_4, G_2.

Kostant--Kumar polynomials distinguish different involutions.

The result applies to Schubert varieties in flag varieties.

Abstract

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.