On the Equivariant Derived Category of Perverse Sheaves

In this paper we extend Beilinson's realization formalism for triangulated categories and filtered triangulated categories to a pseudofunctorial and pseudonatural setting. As a consequence we prove an equivariant version of Beilinson's Theorem: for any algebraic group $G$ over an algebraically closed field $K$ and for any $G$-variety $X$, there is an equivalence of categories $D_G^b(X; \overline{\mathbb{Q}}_{\ell}) \simeq D_G^b(\mathbf{Perv}(X;\overline{\mathbb{Q}}_{\ell}))$ where $\ell$ is an integer prime coprime to the characteristic of $K$. We also show that the equivariant analogues of the other non-$D$-module aspects of Beilinson's Theorem hold in the equivariant case.