TITLE: Reg the motivation behind Lusztig-Vogan bijection
QUESTION [5 upvotes]: Let $G$ be an algebraic group. Choose a Borel subgroup $B$ and
a maximal Torus $T \subset B$. Let $\Lambda$ be the set of weights wrt $T$ and let $\mathfrak{g}$ be the lie algebra of $G$.
Now, consider the following two sets,
1) $\Lambda^+$, the set of dominant weights wrt $B$,
2) The set $N_{o,r}$  of pairs $(e,r)$ (identified upto $\mathfrak{g}$ conjugacy), where e is a nilpotent element in $\mathfrak{g}$ and $r$ is an irreducible representation of the centralizer (in $\mathfrak{g}$) $Z_e$  of a nilpotent element e.
There is a bijective map between the two sets that plays an important
role in the representation theory of $G$ and this is often called the
Lusztig-Vogan bijection,
$\rho_{LV} : \Lambda^+ \rightarrow  N_{o,r}$.
In recent works, this bijection has been studied by Ostrik,
Bezrukovnikov, Chmutova-Ostrik, Achar, Achar-Sommers (Edit: See links to refs below) using various
different tools. My question however pertains to the motivations that
point to the existence of such a bijection in the first place. As I
understand,  the component group $A(O)$ where $O$ is the nilpotent orbit associated to $e$ (under the adjoint action)  and a quotient of the component
group $\overline{A(O)}$ play important roles in the
algorithmic description of this bijection (say for example in determining
the map for certain $h \in \Lambda^+$, where $h$ is the Dynkin element
of a nilpotent orbit in the dual lie algebra).  One of the original motivations
for the existence of such a bijection seems to have emerged from the
study of primitive ideals in the universal enveloping algebra of g. 
My questions are the following :

How does $\overline{A(O)}$ enter the story from the point of view of the
study of primitive ideals ?
Are there other representation theoretic motivations that point to the existence of such a bijection ? Here, I am (somewhat vaguely) counting a motivation to be 'different' if its relation to the theory of primitive ideals is nontrivial. 

[Added in Edit] Refs for some recent works on the bijection (in anti-chronological order) : 

Local systems on nilpotent orbits and weighted Dynkin diagrams (link) - P Achar and E Sommers
Calculating canonical distinguished involutions in the affine Weyl groups (link) - T Chmutova and V Ostrik 
Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone (link) - R Bezrukavnikov

REPLY [4 votes]: There are some conversations on the affine Weyl group cells perspective of the bijection. I guess I can contribute a very little bit on the primitive ideals side of the story, if it is not too late to do so.
My first encounter of the Lusztig's quotient comes from the paper of Barbasch and Vogan in 1985 (here). It is mainly about finding the (g,K)-modules of a fixed infinitesimal character whose annihilators are the 'largest' possible primitive ideal, which Barbasch-Vogan called 'special unipotent' as in the title of the paper.
These special unipotent representations have a lot to do with the ring of regular function of a nilpotent orbit, R[O]. This is first hinted in Section 12 of Vogan's work (here). And Vogan's version of the above bijection is about the structure of R[O], as mentioned in the Achar-Sommers paper. I have carried out some computations on the bijection from this perspective (effectively I can prove a conjecture in the paper). But talking about why Vogan's version of the bijection matches with that of Lusztig's, I think Barbasch and Vogan know much more about it.