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It may be helpful to look at the 1987 paper by Kari Vilonen and his student Ren Mirollo here. They study BGG reciprocity on perverse sheaves in some generality, working over a field. Here the inter …

Since the question involves only classical ideas, it's enough to refer to work of Serre and others explained by Hartshorne in section III.5 of his text on algebraic geometry. In quite a bit of genera …

Modulo the messy bookkeeping sometimes encountered when passing to dual modules, what Kempf does in his paper is to exploit the geometric setting of Cousin complexes in order to find an independent ap …

I'm not quite sure what you are looking for, but Green's work (though combinatorial and influential) was only one of the inputs for the Deligne-Lusztig paper of 1976. It might help for example to …

First I'd comment that there are quite a few questions on MO related to this one, but apparently not quite identical. (It's hard to search the site efficiently.) In any case I won't attempt a detail …

Some added observations on what is likely to be a very complicated problem: The 2010 Selecta Math. paper by Fresse and Melnikov which you mention was posted here, but the published version is somewha …

EDIT: To compensate for my attempted answer, which mainly added further confusion, I'll substitute the following remarks. Note especially that on the algebraic side the confusion starts in the wordin …

Concerning the basic question (a), my first reaction is to be skeptical. Though as you say miracles sometimes do occur in this subject. I don't have a counterexample at my fingertips. Beyond the …

The short answer might be that this viewpoint provides an attractive alternative way to construct the Springer resolution as a special case in a broader geometric framework, following ideas of Kostant …