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 $D_G^b(X;{\bar{\mathbb{Q}}}_{\mathrm{\ell }})\simeq D_G^b(\mathbf{P}\mathbf{e}\mathbf{r}\mathbf{v}(X;{\bar{\mathbb{Q}}}_{\mathrm{\ell }}))$ where $\mathrm{\ell }$ 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.