Kazhdan Lusztig map and Richardson orbits Let $g$ be a simple Lie algebra with Weyl group $W$. Kazhdan and Lusztig defined a map 
$\Phi$:  nilpotent orbits in $g$ $\rightarrow$ conjugacy classes in $W$. 
Let $\eta_p$ be a Richardson orbit associated to a parabolic $p$. 
Question: What is $\Phi(\eta_p)$? 
Here is a natural guess: Let $W_p$ denote the (parabolic) subgroup of $W$ corresponding to $p$. Note that we have a canonical map $i_{p,g}$ from conjugacy classes in $W_p$ to conjugacy classes $W$. Let $c_p$ denote the Coxeter conjugacy class of $W_p$. 
Question: Is it true that $\Phi(\eta_p)=i_{p,g}(c_p)$? 
Remark: Spaltenstein has studied the KL map extensively. However, I could not find the answer to this question in his papers. 
 A: This is a very nice line of thinking! But I think the question, as stated, is imprecise. 
As is correctly pointed out in the question, the KL map takes you from nilpotent orbits in $\mathfrak{g}$ to conjugacy classes in the Weyl group $W(\mathfrak{g})$. However, a given Richardson orbit is not uniquely associated to a single parabolic. In fact, it is not even associated to a unique Levi subalgebra. 
To understand this, it is best to view Richardson orbits as those that are "induced" (in the sense of Lusztig-Spaltenstein) from zero orbits in proper Levi subalgebras. But, a given Richardson orbit can be induced from more than one (non-conjugate) Levi! 
You can however associate a unique "Dixmier sheet" to a pair $(r,l)$, where $r$ is a Richardson Orbit and $l$ is one of the Levis from which it is induced. Now, I think it is a nice question to ask if one can naturally think of an analog of the KL map that takes you from a Dixmier sheet to a fixed conjugacy class of the Weyl Group and if this conjugacy class can be thought of coming from the Coxeter class of the corresponding parabolic sub-Weyl group. 
But, I don't know of any work where something like this is pursued. 
One more potential source of confusion : In the world of Weyl Groups, what is called a parabolic sub-Weyl group really corresponds to what is called a Levi subalgebra in the world of Lie Algebras and not to a "Parabolic" subalgebra. There are far too many Parabolic subalgebras. Even after you fix a Levi, a given Richardson orbit could have multiple "Polarizations" that correspond to choosing a Parabolic with that specific Levi factor. 
As a reference for Richardson Orbits/Sheets/Polarizations, you can consult a paper of de Graaf and Elashvili, "Induced Nilpotent Orbits...". 
