Semiorthogonal decompositions of derived categories of equivariant coherent sheaves

Let $X$ be an algebraic variety with an action of an algebraic group $G$. Suppose that $X$ has a full exceptional collection of sheaves and these sheaves are invariant under the action of the group. We construct a semiorthogonal decomposition of the bounded derived category of $G$-equivariant coherent sheaves on $X$ into components that are equivalent to the derived categories of twisted representations of $G$. If the group is finite or reductive over an algebraically closed field of characteristic 0, this gives a full exceptional collection in the derived equivariant category. We apply our results to particular varieties such as projective spaces, quadrics, Grassmannians and del Pezzo surfaces.