TITLE: Reference for two facts about perverse sheaves on G/B
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: 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.)