This is a question about two seemingly different notions of a left cell in a finite Weyl group and why they are the same. My question arose from reading a paper of W. McGovern titled "Left cells and domino tableaux in classical Weyl groups". I also found Devra Garfinkle's paper "On the classification of primitive ideals for complex classical Lie algebras, I" useful for background.
The first notion comes from the theory of primitive ideals. Let $R_{\lambda}$ be an irreducible highest weight module of $U(\mathfrak{g})$, the universal enveloping algebra of a Lie algebra, with central character $\lambda$. To associate geometric invariants to $R_{\lambda}$, it is interesting to consider the $Ann(R_{\lambda})$, the annihilator of $R_{\lambda}$ in $U(\mathfrak{g})$. An important result of Duflo (see, for ex, the discussion in Intro part of the McGovern paper) says that for any fixed $\lambda$, every primitive ideal $I_{\lambda}$ with central character $\lambda$ is of the form
$I_{\lambda} = Ann(R_{w.\lambda})$,
where
$w.\lambda = -\rho + w\lambda$
for some (not all) $w \in W$, the Weyl group associated to the Lie algebra $\mathfrak{g}$. This is sometimes called the "dotted action" of the Weyl group.
Now, for a fixed central character $\lambda$, asking
$Ann(R_{w.\lambda}) = Ann(R_{w'.\lambda})$
defines an equivalence relation $w \sim w'$ on elements of $W$.
In the above paper, McGovern notes that this equivalence relation is one way to define the notion of a "left cell" in a finite Weyl group. He then observes that this is the same as the definition of a left cell arising in the Kazhdan-Lusztig paper on "Representations of Coxeter groups and Hecke algebras" . In particular, he says that the equivalence of the two definitions is a consequence of KL conjectures (about multiplicities in Verma modules..).
My questions are :
How to see the equivalence of the two definitions from the KL conjectures ?
What is the conceptual* reason for agreement between these two definitions of a left cell ?
*Here, I am requesting any explanation that would give some intuition for why such an agreement should hold. I find this agreement surprising for many reasons. One of them is that the KL theory (of cells) works only with the Coxeter Group structure while the primitive ideal theory remembers more data (like an underlying root system). For example, (I think) the classification of primitive ideals in $B_n$ and $C_n$ would have some differences but $W(B_n)$ and $W(C_n)$ are isomorphic as Coxeter groups and hence have identical KL theory.
