The minimal degeneration singularities in the affine Grassmannians of simple simply laced algebraic groups are determined to be either Kleinian singularities of type A or closures of minimal nilpotent orbits. The singularities for non-simply laced types are studied by intersection cohomology and equivariant Chow group methods.
"The minimal degeneration singularities in the affine Grassmannians." Duke Math. J. 126 (2) 233 - 249, 15 February 2005. https://doi.org/10.1215/S0012-7094-04-12622-3