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I wonder whether there is a reference for the following two things:

The Grothendieck group of B-equivariant semisimple? perverse sheaves on $G/B$ is the Hecke-algebra.

The category of B-equivariant perverse sheaves on $G/B$ is equivalent to those modules of category $\mathcal O$, where the center acts trivial.

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3 Answers

up vote 8 down vote accepted

Both of these are facts which developed gradually over the course of several papers, so it's hard to give a definitive reference.

For the first, all the calculations one would need in order to establish this fact are done in, for example, Springer's Quelques applications de la cohomologie d’intersection, but this isn't stated as a theorem. If you're just looking for somewhere in the literature to cite, this is stated as Theorem 4 in The geometry of Markov traces by myself and Geordie Williamson. EDIT: And Bugs is completely right; you need to say "mixed" here, or you just get the group algebra of the Weyl group.

For the second, I would say $N$-equivariant, or Schubert smooth, rather than $B$-equivariant, since the $B$-equivariant derived category is the wrong thing (this is like the difference between category $\mathcal O$ and translation functors on it). You should also be careful about what category you're talking about; you don't want the center to act trivially, but nilpotently. The easiest reference I know of is Koszul Duality Patterns in Representation Theory by Beilinson, Ginzburg and Soergel, Proposition 3.5.2, though the theorem is older, going back to Soergel's Habilitationsschrift.

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Did he say $B$-equivariant derived category or derived category of $B$-equivariant sheaves? The former is, indeed, wonky but the latter seems to be right... –  Bugs Bunny Dec 17 '10 at 9:16
    
Note I didn't say it was the wrong thing, I said it was terminogy I prefer to avoid. It's a bad habit which can lead to trouble later of you think about this stuff seriously. –  Ben Webster Dec 17 '10 at 18:53
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You need to add mixed to your first statement...

If you need to refer, I'd say refer to the original Brylinski-Kashiwara proof of Kazhdan-Lusztig conjecture and unpublished Bernstein's lecture notes on d-modules.

If you want to learn the stuff, start with Bernstein's lecture notes and continue onto unpublished Milicic's book

And Koszul Duality Patterns recommended by Ben is an excellent source as well. Somehow it takes you to the same material by a back-door (I did not read other Ben's references...)

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Actually, the Brylinski-Kashiwara paper doesn't prove an equivalence between Schubert smooth D-modules and what representation theorists usually call category O. The modules that show up as sections of such D-modules aren't always in category O, since they don't always have a weight decomposition, and they miss many modules of category O, since they must have trivial action of the center (not just nilpotent). Soergel proved later on that there is an equivalence to category O, but it's a non-obvious functor. –  Ben Webster Dec 17 '10 at 19:03
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A couple of things (I meant to just have this as a comment but it got too long): I think with regard to the first fact, a lot of people realised it all around the same time -- the article by Springer which Ben mentions does all the calculations needed, as he says, but doesn't state the result, while an article not long after by Lusztig and Vogan in 1983 states the result at the end, and it's mentioned elsewhere in the literature that Brylinski also proved it (and I can't believe Beilinson and Bernstein didn't know also). It's now, however, textbook stuff thanks to the book by Hotta Tanisaki and Takeuchi.

As to the second, I think Ben is overcomplicating things a little. If I say B-equivariant perverse sheaves, that will force the $\mathfrak h$-semisimplicity, which is what you need e beyond say the equivalence which is proved in the paper of Kashiwara-Brylinski. Of course Ben is right to point out that more care needs to be taken with the derived categories, but there are other references for that -- e.g. Bernstein and Lunts paper. The other classic reference that should be mentioned here is the paper of Beilinson and Bernstein on the Jantzen filtration, which discusses a lot of the issues related to central characters (e.g. considering the centre acting trivially or nilpotently etc.)

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