Let $\mathfrak{g}$ be a complex semisimple Lie algebra. This question concerns three classical objects of representation theory: the two-sided Kazhdan-Lusztig cells of the Weyl group $W$ of $\mathfrak{g}$, the irreducible representations of $W$ which are special in the sense of Lusztig, and the special nilpotent orbits of $\mathfrak{g}$. All these objects form finite sets, and we have the following diagram of bijective correspondences

[![main diagram][1]][1]

which I'll discuss a bit more in a minute. My question is, *does this diagram commute*? Surely the answer is yes, but I can't see a proof of this in the literature.

Here are some more details on the objects involved and their relations to one another:

 - The KL cells are the equivalence classes of a certain equivalence relation on $W$, which is defined using the Hecke algebra of $W$ and its Kazhdan-Lusztig basis. To every two-sided cell $\mathbf{c}$, Lusztig attached a $W$-bimodule $[\mathbf{c}]$; two irreducibles $E, E'\in \mathrm{Irr}(W)$ are said to belong to the same *family* if they both appear as irreducible components of $[\mathbf{c}]$. Lusztig's families partition the set $\mathrm{Irr}(W)$. Moreover, each family contains a unique *special* representation; this means equality is achieved in a certain numerical inequality $a_E \leq b_E$. This yields the bijection labelled 'Lusztig' in the diagram above.
- A nilpotent orbit $\mathbb{O}$ is orbit of the nilpotent cone $\mathcal{N}$ of $\mathfrak{g}$ under the action of its adjoint group. Fix $e \in \mathbb{O}$, let $A(e)$ denote the component group of the centraliser of $e$, and consider the fibre $\mathcal{B}_e$ of $e$ in the Springer resolution $T^* \mathcal{B} \to \mathcal{N}$ (here  $\mathcal{B}$ denotes the flag variety of Borel subalgebras). Springer constructed an action of $W \times A(e)$ on the cohomology space $\mathrm{Sp}(\mathbb{O}) := H^{2d}(\mathcal{B}_e, \mathbb{C})$ (here $d = \dim \mathcal{B}_e$), obtaining a decomposition $$ \mathrm{Sp}(\mathbb{O}) = \oplus_{E \in \mathrm{Irr}(W)} E \otimes V_E, $$
where each $V_E$ is a complex finite-dimensional irreducible representation of $A(e)$. The trivial representation $V_E = \mathbb{C}$ always occurs exactly once in this decomposition; if the corresponding $W$-module $E$ is special in the above sense, then we say that the orbit $\mathbb{O}$ is special. This yields the bijection labelled 'Springer' in the diagram above.
- Let $\mathcal{U}$ denote the universal enveloping algebra of $\mathfrak{g}$, and consider the set $\mathrm{Prim}(\mathcal{U})$ of primitive ideals of $\mathcal{U}$ (i.e. two-sided ideals which are annihilators of simple left $\mathcal{U}$-modules) with trivial central character.  Duflo proved that every $J \in \mathrm{Prim}(\mathcal{U})$ can be written as the annihilator of a simple highest weight module $L_w$, so that we obtain a surjective map 
$$W \twoheadrightarrow \mathrm{Prim}(\mathcal{U}), \quad w \mapsto \mathrm{Ann} (L_w).$$
By work of [Barbasch-Vogan][2], the fibres of this map are exactly the left KL cells of $W$. Moreover, if we let $\mathrm{Prim}_{\mathbb{O}}(\mathcal{U})$ denote the set of $J \in \mathrm{Prim}(\mathcal{U})$ such that the associated variety of $\mathcal{U} / J$ equals $\overline{\mathbb{O}}$, then this set is nonempty iff $\mathbb{O}$ is special, and in this case its preimage under Duflo's map is a two-sided cell of $W$. This yields the bijection labelled 'Barbasch-Vogan' in the diagram above.

I'm especially interested in the case where $\mathbb{O}$ is the subregular orbit and $\mathfrak{g}$ is simple of type ADE. Then the corresponding Springer representation $\mathrm{Sp}(\mathbb{O})$ is irreducible (since $A(e)$ is trivial), and is isomorphic to the reflection representation of $W$. On the other hand, the two-sided cell $\mathbf{c}$ corresponding to the reflection representation via Lusztig's construction can be characterised as the set of elements of $W$ having a unique reduced decomposition; equivalently, $\mathbf{c} = \sqcup_{s\in S} \sigma_s $, where $\sigma_s$ is the left cell containing the simple reflection $s$; equivalently, $\mathbf{c} = a^{-1}(1)$, where $a : W \to \mathbb{Z}_{\geq 0}$ is the function defined by Lusztig. I want to be sure that this $\mathbf{c}$ coincides with the preimage of $\mathrm{Prim}_{\mathbb{O}}(\mathcal{U})$ under the Duflo map. 


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EDIT: Here is a rough attempt at an answer, which makes use of the 'cell category' $\mathrm{HC}^{ss}_{\mathbb{O}}(\mathcal{U})$ associated to a special nilpotent orbit $\mathbb{O}$. This category consists of the semisimple objects in a certain quotient category: the quotient of the category of Harish-Chandra bimodules $M$ with trivial central character and whose associated variety $V(M)$ is contained in $\overline{\mathbb{O}}$, by the subcategory of objects such that $V(M) \subset \partial \overline{\mathbb{O}}$. In [this paper][3], Bezrukavnikov, Finkelberg and Ostrik prove that $\mathrm{HC}^{ss}_{\mathbb{O}}(\mathcal{U})$ is tensor equivalent to Lusztig’s categorification of the block $J^{\mathbf{c}}$ of the asymptotic Hecke algebra $J$, where $\mathbf{c}$ is the two-sided cell equal to the preimage of $\mathrm{Prim}_{\mathbb{O}}(\mathcal{U}) = \{ J_1, \ldots, J_n\}$ under Duflo's map. We remark that $\mathrm{HC}^{ss}_{\mathbb{O}}(\mathcal{U})$ is a multi-fusion category (essentially a rigid tensor category consisting of semisimple objects), with simple objects represented by the primitive quotients $\mathcal{U} / J_1, \ldots, \mathcal{U} / J_n$. For a great exposition of the cell category, see [this paper][4] of Ostrik.

Let $E$ be the special representation of $W$ attached to $\mathbb{O}$ via the Springer correspondence. We will interpret $E \subset \mathbb{C}[\mathfrak{h}^*]$ as a Goldie-rank representation; if $p_i$ denotes the Goldie-rank polynomial of  $\mathcal{U} / J_i$, then $\{p_1, \ldots, p_n \}$ is a basis for $E$. Let $E_\infty$ be the simple $J_{\mathbb{C}}$-module corresponding to $E$ via Lusztig’s map $\mathbb{C}[W] \cong J_{\mathbb{C}}$. Then $E_\infty$ is a simple $J^{\mathbf{c}’}_{\mathbb{C}}$-module for a unique two-sided cell $\mathbf{c}’$. We aim to show $\mathbf{c} = \mathbf{c}’$ by showing $E_\infty$ is a simple $J^{\mathbf{c}}_{\mathbb{C}}$-module. 

Recall that $K(\mathrm{HC}^{ss}_{\mathbb{O}}(\mathcal{U})) \cong J^{\mathbf{c}}$. It should be enough to make $E$ into a $K(\mathrm{HC}^{ss}_{\mathbb{O}}(\mathcal{U}))$-module in a way which is compatible with the $\mathbb{C}[W]$-module structure it already possesses. There seems to be a natural way to do this: if we let $\mathcal{U}/J_i$ act diagonally on the basis elements $p_j \mapsto \delta_{ij} p_j$, then this defines an algebra homomorphism $K(\mathrm{HC}^{ss}_{\mathbb{O}}(\mathcal{U})) \to \mathrm{End}_{\mathbb{C}}(E)$, since the $\mathcal{U}/J_i$ form a set of mutually orthogonal idempotents for $\mathrm{HC}^{ss}_{\mathbb{O}}(\mathcal{U})$. To complete the proof it remains to check that this action is compatible with the canonical map $\mathbb{C}[W] \cong J_{\mathbb{C}} \twoheadrightarrow J^{\mathbf{c}}_{\mathbb{C}}$, which I won't attempt to do right now.



  [3]: https://arxiv.org/pdf/0902.1493.pdf
  [4]: https://arxiv.org/abs/1404.6575

  [1]: https://i.sstatic.net/bT5dV.png
  [2]: https://www.sciencedirect.com/science/article/pii/0021869383900066?via%3Dihub