This is a follow-up to a recent question asked by Peter Crooks here. The answer by Ben Webster includes a helpful link to the corrected arXiv version of Baohua Fu's 2003 Invent. Math. paper *Symplectic resolutions for nilpotent orbits*.

Most of this literature is unfamiliar to me, so I may be overlooking something. I've encountered nilpotent orbits mainly in connection with various types of representation theory (sometimes in good prime characteristic, where most properties of the orbits are the same as over $\mathbb{C}$). At this point I'm still confused about some details, such as:

Are there Richardson orbits whose closures fail to have a symplectic resolution (and if so, what is the lowest rank Lie algebra in which an example appears)?

EDIT: Here I'm using shorthand to avoid normality questions: read "for which the normalizations of their closures fail to have ...? (Apparently the sorting out of normal orbits isn't complete yet for some exceptional types.)

As Fu notes in Prop. 3.16, it follows from the main theorem of the paper that a nilpotent orbit whose closure admits a symplectic resolution must be *Richardson* (intersecting the nilradical of some parabolic subalgebra in a dense orbit). In turn a reviewer states: "But the converse is not always true." I don't see direct evidence of that in Fu's paper. Here the simple Lie algebras are studied case-by-case: all orbits in type $A_n$ are Richardson, with trivial component groups, forcing their closures to have symplectic resolutions. In types $G_2, F_4, E_6$, all Richardson orbits also have trivial component groups, whereas a few such orbits in types $E_7, E_8$ have component groups of order 2 and are left unsettled in the paper. (These cases were later treated geometrically here.)

The discussion of types $B_n, C_n, D_n$ leaves me somewhat confused, since the explicit examples mentioned between Prop. 3.21 and Prop. 3.22 aren't Richardson orbits. This prompts the question above.

ADDED: Fu reduces the problem (for a Richardson orbit) to the question of whether or not there exists a parabolic $P$ defining the orbit for which $N(P)=1$. This is the index in the full component group in $G$ (topologically, fundamental group) of the component group in $P$ of an orbit element $X$. (Here "component group" means the group $C_G(X)/C_G(X)^\circ$.) It's not clear how to compute $N(P)$ in all cases, which may be why Fu gave up on the leftover cases in types $E_7,E_8$. I'm wondering about the naive (presumably false?) statement that an orbit closure has a symplectic resolution iff the orbit is Richardson. What bothers me is that Fu seems to mention only examples of non-Richardson orbits such as minimal orbits in types other than $A_n$, etc.