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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 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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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.

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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