TITLE: the affine coordinate ring of orbit closures in the ordinary nilpotent cone
QUESTION [5 upvotes]: Given a partition $\lambda$ of $n$, consider the orbit closure $\overline{ \mathcal{O}_{\lambda}}$ of the nilpotent orbit corresponding to that partition. My question, is how to explicitly construct the affine coordinate ring of the (singular) variety that is the closure of this orbit?
My second question, is the same but for the orbit closure of an orbit in the enhanced nilpotent cone (see, for instance, Achar-Henderson's ``Orbit closures in the enhanced nilpotent cone"). How would you explicitly find equations generating the ideal killing that variety in this case? 
At least the first question is probably well-known and an exercise in combinatorics/linear algebra, but I am having trouble finding a reference - the closest I could find is a paper describing the Springer correspondence for types other than A via the definition of the Weyl group acting on the weight $0$ space, but I couldn't find the answer there.

REPLY [6 votes]: Weyman has calculated generators of the radical ideal of $\overline{\mathcal{O}_{\lambda}}$ for every $\lambda$. See "The equations of conjugacy classes of nilpotent matrices", 
Invent. Math. 98 (1989), no. 2, 229–245. In particular, see Theorem 4.6 and the definitions which precede it.