Equivariant derived category of a complete symmetric variety

Abstract:

Let $G$ be a complex algebraic semi-simple adjoint group and $X$ a smooth complete symmetric $G$-variety. Let $L= \bigoplus _\alpha L_\alpha$ be the direct sum of all irreducible $G$-equivariant intersection cohomology complexes on $X$, and let $\mathcal E= \operatorname {Ext}^\cdot _{\mathrm {D}_G(X)}(L,L)$ be the extension algebra of $L$, computed in the $G$-equivariant derived category of $X$. We considered $\mathcal E$ as a dg-algebra with differential $d_\mathcal E =0$, and the $\mathcal E_\alpha = \operatorname {Ext}^\cdot _{\mathrm {D}_G(X)}(L,L_\alpha )$ as $\mathcal E$-dg-modules. We show that the bounded equivariant derived category of sheaves of $\mathbf {C}$-vector spaces on $X$ is equivalent to $\mathrm {D}_\mathcal E\langle \mathcal E_\alpha \rangle$, the subcategory of the derived category of $\mathcal E$-dg-modules generated by the $\mathcal E_\alpha$.