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

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