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