Cells in affine Weyl groups This may sound like a very general question, which pretty much reflects my ignorance on the subject.
In the case of Weyl groups $W$, there is a notion of left/right/double cells, which is roughly some partitions on the elements of $W$. It turns out that these cells afford representations of $W$ (or $W \times W$ in the case of double cells).
The structure of these cells are well-known, for example in the case of $W = S_n$, the partition of cells is related to the Robinson-Schensted algorithm. 
There are applications of cells in other aspects of mathematics. Due to my ignorance of the subject, I will simply refer what I mean by 'other aspects' to the notes here. 
On the other hand, Lusztig also defined 'cells' in affine Weyl groups in a series of papers under the title (Cells in affine Weyl groups), which I have literally zero knowledge about. Here is a couple of questions I am particularly interested in:
1) I read from Sommers-Gunnells (here p.7) that the double cells in affine Weyl group are parametrized by nilpotent orbits in the Langlands dual. This, to my knowledge, is quite different from the case of double cells in Weyl group, whose double cells are parametrized by special orbits in the Langlands dual (correct me if I am wrong). Can anyone give us some intuition on how the parameterization in the affine Weyl group case works?
2) There is a notion of canonical left cell in each double cell of affine Weyl group (here). Are there any results on how these canonical left cells are computed?
 A: You are asking a number of related questions here, most of which require more reading of Lusztig's papers.   See the reference list in my conference paper here, for example.  But note first that the notions about cells you mention are defined quite generally for Coxeter groups in the landmark 1979 paper by Kazhdan and Lusztig here.    What is special about affine or finite Coxeter groups is the precise detail Lusztig later worked out.   For example, in the fourth part of his series of papers on cells in affine Weyl groups, he arrived at a proof of his conjectured bijection between 2-sided cells and nilpotent orbits in the Lie algebra of a Langlands dual algebraic group of the relevant type.  [But it is very hard to get an intuitive feeling for this bijection.   As far as I know, there is not yet an alternative to Lusztig's complicated proof, which uses virtually all possible tools.  So this part of the subject remains challenging.]
There is still no proof of an older 1983 conjecture by Lusztig on the precise number of left cells in a 2-sided cell.    This and other such combinatorial matters are related to unsolved problems in representation theory.   Anyway, the precise relationship between cells in affine Weyl groups and finite Weyl groups has been worked out in Lusztig's papers and is nicely illustrated in the rank 2 cases in his Japanese conference paper. (See also the photo of Lusztig in his specially made $G_2$ t-shirt in the paper by Gunnells here.)   
Concerning canonical left cells, there has been further computational work by Chmutova and Ostrik in a 2002 paper here.  Gunnells and others have gone on to study further the difficult problem of characterizing the location of "distinguished involutions" in all left cells, which I suspect is closely related to modular representations of the corresponding Lie algebras. 
ADDED: To comment a little further, the occurrence here of the Langlands dual group is perhaps mysterioous, but for the reflection groups and cells it shows up mainly in the interchange of Lie types $B_\ell$ and $C_\ell$.   The embedding of a Weyl group into an affine Weyl group is less affeccted, since Weyl groups of these types are isomorphic.   But the dual version of an affine Weyl group, with its translation lattice expanded by a prime factor $p$, is crucial in characteristic $p$ representation theory of the algebraic groups involved (as Verma first noticed).
From Lusztig's work (especially his definition of "special" nilpotent orbits), one sees that a two-sided cell of the Langlands dual affine Weyl group meets a (unique) two-sided cell of the finite Weyl group precisely when the corresponding nilpotent orbit is special.    In type $A_\ell$ where $W = S_{\ell+1}$, all orbits are special, but otherwise not.  This adds a further layer of interest to the kind of parametrization you ask for.   Special orbits have a Lusztig-Spaltenstein duality, for type $A_\ell$ just the transpose involution on partitions of $\ell+1$.   This leads further, to Lusztig's special pieces of the nilpotent variety, which people now understand better in terms of an "exotic" version of the variety: see for example here. 
Some short unpublished notes (in a more general context, where the finite Weyl group appears as one of the parabolic subgroups of an affine Weyl group) are here, but don't deal with the geometric side.
