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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 dâ€™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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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 dâ€™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.

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.