  For a simply-connected simple algebraic group $G$ over $\C$, we exhibit a
subvariety of its affine Grassmannian that is closely related to the nilpotent
cone of $G$, generalizing a well-known fact about $GL_n$. Using this variety,
we construct a sheaf-theoretic functor that, when combined with the geometric
Satake equivalence and the Springer correspondence, leads to a geometric
explanation for a number of known facts (mostly due to Broer and Reeder) about
small representations of the dual group.
