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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.

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    $\begingroup$ As Aswin points out, the question is not well-formulated. Type $A_\ell$ may be misleading, since all orbits then are Richardson and the KL map is bijective. But there is an inevitable problem when two root lengths occur, since the subregular nilpotent orbit is then the Richardson orbit coming from the minimal parabolics; but their Coxeter elements need not be conjugate in $W$, as seen first in type $B_2 (= C_2)$. (Also, a more standard notation for a parabolic subalgebra is $\mathfrak{p}$ relative to a set $I$ of simple reflections.) $\endgroup$
    – Jim Humphreys
    Jan 5, 2017 at 19:26
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    $\begingroup$ Concerning your first question, it's natural to start with a closer look at the simple types as well as at special choices of nilpotent orbits. Kazhdan-Lusztig already work out some examples in their paper, e.g., the regular orbit, where you do get a Coxeter class: see their 9.12(a). $\endgroup$
    – Jim Humphreys
    Jan 5, 2017 at 19:32

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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...".

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  • $\begingroup$ Thanks Aswin. Your comment made me realise that the answer to my second question is "no". According to the book of Colingwood-McGovern, p 127, if we consider the Levis in F_4 which have semisimple types $A_2\times A_1$ and $B_2$, then the corresponding Richardson orbits coincide. However, the images of the Coxeter classes of these Levis inside conjugacy classes of $W$ do not coincide. Indeed, the Coxeter numbers are $6$ and $4$, respectively and one has non-canonical inclusions $W_l\rightarrow W$ and $W_{l'}\rightarrow W$ (which give rise to canonical maps on the level of conjugacy classes). $\endgroup$
    – Dr. Evil
    Nov 22, 2016 at 0:49
  • $\begingroup$ Nice Dr. Evil, good to have it clarified! $\endgroup$
    – Aswin
    Nov 24, 2016 at 10:04

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