Tangent spaces of spherical Schubert varieties and a counterexample to the reducedness conjecture

Abstract: Given a simply-connected simple algebraic group $G$, we determine the tangent spaces of the Finkerberg-Mirković Schubert scheme in the affine Grassmannian of $G$ associated to the quasi-miniscule coweight. As a consequence, we exhibit a non-reduced Finkerberg-Mirković Schubert scheme when $G$ is of type $E_8$.