Reg the motivation behind Lusztig-Vogan bijection

Question

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

Answer 1

Like Jay, I don't see any reasonable way to address all parts of your wide-ranging question. You are looking at the intersection of numerous lines of research, motivated in different ways for different people. For myself, the primary motivation comes indirectly from modular representations of Lie algebras attached to simple algebraic groups in (good) prime characteristic. Here the basic machinery of nilpotent orbits and component groups is essentially the same as in the classical work over $\mathbb{C}$ which you are implicitly referring to.

There are many ingredients here, suggested in the organization of a conference note I wrote a decade ago here. It includes a more extensive list of related papers including those already mentioned in the question. (All of this was heavily influenced by conversations I had with Roman Bezrukavnikov, but the program sketched remains speculative.)

Primitive ideals are almost certainly lurking in the background of what I've written down, as well as in other interpretations of the L-V bijection. In prime characteristic A. Premet has intriguing ideas in some of his papers about reduction mod $p$ of certain primitive ideals in a characteristic 0 enveloping algebra, which are directly relevant to the modular representation theory. But many of the questions raised, starting with Lusztig's series of papers in the 1980s on cells in affine Weyl groups, remain only partly answered. At least the literature shows a range of motivations and applications involving representation theory, along with a lot of basic machinery. But for Lusztig's canonical quotient of the component group (with your bar notation), you really need to look at the papers by Achar and Sommers. This literature goes in many directions including the Springer correspondence, in spite of having some unity under the surface.

One thing I should emphasize is that Lusztig's bijection requires a transition to an affine Weyl group attached to the Langlands dual group. (This already became part of the modular theory in Verma's work in the early 1970s.) One of Lusztig's basic ideas is to pass from nilpotent orbits to 2-sided cells in the dual type of affine Weyl group.

Finally, it helps to start with the simplest example of this complicated set-up, where $G = \mathrm{SL}_2$. In the Lie algebra you have two nilpotent orbits: the zero orbit and the regular orbit. So the centralizers (in the adjoint group) are respectively the entire group and a unipotent group having only the trivial irreducible representation. The latter pair should be associated with the zero weight (in the root lattice!), whereas the infinitely many irredudible representations of the adjoint group $\mathrm{PGL}_2$ are usually indexed by the integers $\{0,2,4, \dots\}$. But in the bijection here you need to shift thse by 2 ($=2\rho$). This may seem arbitrary but does make sense in the cell picture and the modular theory.

Answer 2

This isn't even vaguely an answer to your question but is more of a clarifying remark concerning the canonical quotient. Throughout I will write [Lus84] for Lusztig's orange book "Characters of reductive groups over a finite field", Princeton University Press, 1984.

In what follows I will assume that $\mathbf{G}$ is a connected reductive algebraic group over an algebraic closure $\overline{\mathbb{F}_p}$ where $p$ is a good prime for $\mathbf{G}$. Furthermore, I will denote by $F : \mathbf{G} \to \mathbf{G}$ a Frobenius endomorphism and $G = \mathbf{G}^F$ the corresponding finite reductive group.

Let us denote by $\mathcal{E}(G,1)$ Lusztig's set of unipotent characters. These are defined to be all irreducible characters occuring in a Deligne–Lusztig virtual character $R_{\mathbf{T}}^{\mathbf{G}}(1)$ where $\mathbf{T}$ is an $F$-stable maximal torus of $\mathbf{G}$ and 1 denotes the trivial character. In [Lus84] Lusztig has defined a partitioning of $\mathcal{E}(G,1)$ into what he calls families. These are naturally in bijection with the 2-sided cells of the corresponding Weyl group of $\mathbf{G}$.

Now, to each family $\mathcal{F} \subseteq \mathcal{E}(G,1)$ Lusztig has defined on a case by case basis (see Chapter 4 of [Lus84]) a small finite group $\mathcal{G_F}$. This group, and its irreducible characters, plays an important role in the representation theory of $G$. In particular, this is used to not only give a labelling to the irreducible characters in $\mathcal{F}$ but also to determine the multiplicity of $\chi \in \mathcal{F}$ in the $R_{\mathbf{T}}^{\mathbf{G}}(1)$'s. However one would like a more natural interpretation for this finite group.

Using Lusztig–Macdonald–Spaltenstein induction and the Springer correspondence Lusztig has associated to every family $\mathcal{F} \subseteq \mathcal{E}(G,1)$ an $F$-stable unipotent class $\mathcal{O}_{\mathcal{F}}$ of $\mathbf{G}$. This turns out to be the unipotent support of the characters in $\mathcal{F}$ (see Lusztig, "A unipotent support for irreducible representations", Adv. Math., 1992). What Lusztig saw (see Chapter 14 of [Lus84]) was that the small finite group $\mathcal{G_F}$ is not exactly the component group $A(\mathcal{O}_{\mathcal{F}})$ but it is a quotient of this group, namely Lusztig's canonical quotient group $\overline{A}(\mathcal{O}_{\mathcal{F}})$.

This is quite vague but I hope it gives a bit more of an idea for the origins of Lusztig's canonical quotient.

Answer 3

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.