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The Kazhdan Lusztig map gives a correspondence between conjugacy classes of Weyl groups and nilpotent orbits. So, let $w$ be a conjugacy class and $N$ be the nilpotent orbit it gets mapped to under the KL map. My question is

Is there a simple relationship between $n(w)$ and $dim(N)$ ? (alternatively, is this data obtainable from the combinatorial information contained in the KL polynomials ?).

Notation : $n(w)$ is the number of elements in the conjugacy class and $dim(N)$ is the complex dimension of the nilpotent orbit.

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1 Answer 1

up vote 6 down vote accepted

The question itself is not precisely enough stated (or documented) to permit a real answer, I think. It's an old problem to specify some "natural" correspondence between conjugacy classes in a Weyl group and nilpotent orbits in a related semisimple complex Lie algebra (or perhaps unipotent classes in a corresponding algebraic group). Here the special linear Lie algebra, with a corresponding symmetric group as Weyl group, offers a clearcut but oversimplified example: use the common partition labelling of both nilpotent orbits (via Jordan form) and classes in the symmetric group. But here the orbit dimensions and the sizes of the classes don't seem to be correlated in an obvious way: consider for example the partitions paired by transposing, or the natural partial ordering of partitions.

In general the number of orbits will be less than the number of classes (or characters) in this situation, so it gets more subtle. (The Springer correspondence illustrates this subtlety.) In his early work, Roger Carter did make a systematic but complicated attempt to relate the two pictures. The work of Kazhdan and Lusztig led them to propose their own map from nilpotent orbits to conjugacy classes in $W$: see Fixed point varieties on affine flag manifolds, Israel J. Math. 62 (1988), section 9. Going in this direction, they suggested that their map should be injective for all types.

This is already complicated enough to set up, but in a more recent series of papers (posted on the arXiv) Lusztig has continued his search for the right way to look at all of this. For example, he tries working in the other direction, expecting to find a surjective map. I'm not enough of an expert on what's in all these papers to sort out whether dimensions of orbits and sizes of classes correlate at all. But getting the "right" map in either direction is not a triviality and needs to be specified in the question asked. Along with some indication of which special cases you've already looked at.

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I think Aswin is interested in the following: given the specific map from nilpotent orbits to conjugacy classes in W defined by Kazhdan-Lusztig in the paper you helpfully cite (if I recall correctly, consider a generic arc through a given nilpotent orbit and assign the monodromy data of the corresponding regular semisimple element over the corresponding fraction field), what is the size of the resulting conjugacy class as a function of the nilpotent orbit? –  David Ben-Zvi Sep 5 '12 at 2:15
    
@David: I'm basically requesting a precise formulation. In particular, the direction of the "Kazhdan-Lusztig" map is fuzzy: conjugacy classes to nilpotent orbits, or the other way? Anyway, the dimensions of nilpotent orbits don't seem to be relevant here relative to the sizes of conjugacy classes, unless I'm missing something. –  Jim Humphreys Sep 6 '12 at 11:26
    
@David Thanks for making my question more precise. @Jim Humphreys Thanks for the answer. I did have the lie algebra $sl_n$ in mind when I posed the question (to avoid subtleties of the map not being bijective) and should have stated it explicitly. As you point out, the orbit dimensions and the number of elements in the classes don't seem to be correlated in any special way. My question was basically to know if there actually was a connection known in the literature that I had missed. –  Aswin Sep 9 '12 at 18:21
    
@Aswin: It's probably reassuring to check the small case of Lie type $G_2$, where the five nilpotent orbits have dimensions 0, 6, 8, 10, 12, but the six conjugacy classes in the Weyl group (dihedral of order 12) have sizes 1, 1, 2, 2, 3, 3 (or something like that). –  Jim Humphreys Sep 10 '12 at 20:11

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