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.



Reference for two facts about perverse sheaves on G/B
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](http://archive.numdam.org/ARCHIVE/SB/SB_1981-1982__24_/SB_1981-1982__24__249_0/SB_1981-1982__24__249_0.pdf), 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](http://arxiv.org/pdf/0911.4494.pdf) 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](http://www.ams.org/journals/jams/1996-9-02/S0894-0347-96-00192-0/) by Beilinson, Ginzburg and Soergel, Proposition 3.5.2, though the theorem is older, going back to Soergel's Habilitationsschrift.


