On a reducedness conjecture for spherical Schubert varieties and slices   in the affine Grassmannian
  We study spherical Schubert varieties in the affine Grassmannian. These Schubert varieties have a natural conjectural modular description due to Finkelberg-Mirkovi\'c. This modular description is easily seen to be set-theoretically correct, but it is not obviously scheme-theoretically correct. We prove that this modular description is correct in many cases. We also link this modular description to the reducedness conjecture from Kamnitzer-Webster-Weekes-Yacobi for tranverse slices in the affine Grassmannian. 
Joel Kamnitzer, Dinakar Muthiah, Alex Weekes
math.RT math.AG