Semiorthogonal decompositions of derived categories of equivariant coherent sheaves

Let X be an algebraic variety with an action of an algebraic group G. Suppose X has a full exceptional collection of sheaves, and these sheaves are invariant under the action of the group. We construct a semiorthogonal decomposition of bounded derived category of G-equivariant coherent sheaves on X into components, equivalent to derived categories of twisted representations of the group. If the group is finite or reductive over the algebraically closed field of zero characteristic, this gives a full exceptional collection in the derived equivariant category. We apply our results to particular varieties such as projective spaces, quadrics, Grassmanians and Del Pezzo surfaces. Introduction. Let X be an algebraic variety over the field k with an action of an algebraic group G. In this paper we investigate D b (coh G (X)) — the derived category of G-equivariant coherent sheaves on X. We prove that under some conditions the category D b (coh G (X)) admits a semiorthogonal decomposition into subcategories, equivalent to derived categories of