# Kazhdan Lusztig Map and conjugacy classes of Weyl groups.

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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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.
@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