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. 
 A: 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.
