Tangent spaces of spherical Schubert varieties and counterexamples to the reducedness conjecture

TL;DR

This paper computes tangent spaces of certain Schubert schemes in the affine Grassmannian and provides counterexamples to the reducedness conjecture for groups of types E6, E7, and E8.

Contribution

It explicitly determines tangent spaces at the base point and identifies non-reduced schemes in specific exceptional types.

Findings

Explicit tangent space descriptions for Finkelberg-Mirković Schubert schemes.

Counterexamples to the reducedness conjecture in types E6, E7, and E8.

Non-reduced schemes are exhibited for these exceptional types.

Abstract

Given a simply-connected simple algebraic group $G$, we determine the tangent space of any Finkelberg-Mirkovi\'c Schubert scheme at the base point of the affine Grassmannian of $G$. As a consequence, we exhibit non-reduced Finkelberg-Mirkovi\'c Schubert schemes when $G$ is of type $E_6,E_7$ and $E_8$.