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@MarkWildon I think its more a question of convention than anything else. If I write a sum over all representation ring of symmetric group that means that an element is a finite sum while if I define the ring of symmetriic functions as a limit i get series with an infinite number of terms. There are probably two ways to fix this, one is to take the product in my original definition, the other is taking some kind of colimit in the definition of symmetric functions (as is done in the wikipedia article on symmetric functions).
Could you elaborate please on the last point. What does it mean that they are dual to the representations of the quantum group? What kind of object is it? Is it a hopf algebra over $\mathbb{Z}[q]$ specializing at $q=1$ to the hopf algebra of the general linear group?
@MarcHoyois Could you help me understand, or point to a reference, what does "geometric" mean for a motive. I familiar with the general construction of taking nisnevich sheaves and localizing by $A^1$-equivalences and stabilizing then rationalizing. What do I need to do from this point to get "geometric motives? Persumably those should correspond to the numerical equivalence relation, how is it encoded in this language?
Sato's construction of the microlocalization functor which goes from $Sh(X)$ to $Sh(T^{*}X)$ uses in an essential way the Laplace transform between sheaves on the tangent bundle and sheaves on the cotangent bundle. Notice that laplace transform is kind of a fourier transform for the real setting.
Yes, I understand now, of course you are correct, sorry. This covers the proper-smooth case. Do you know a simple example of a surface singularity with a divergent formal deformation? Just to make this answer complete
I changed the question in hope that it would be more precise so that I could understand the answer. I'm not sure how an elliptic curve can have a divergent $j$-invariant if it comes from a formal deformation. In particular I think both singular affine curves and complete non-singular curves can't be counterexamples. Could you be more precise please? Sorry about the abrupt edit of the question.
You might be interested in the following recent paper by D. Rumynin: arxiv.org/abs/1811.08612v1 in which he proves that any smooth rational projective $D$-affine variety is a generalized flag variety (in char 0).
