The decomposition theorem and the intersection cohomology of quotients in algebraic geometry

Suppose a connected reductive complex algebraic group G acts linearly on an irreducible complex projective variety X. We prove that if 1→N→G→H→1is a short exact sequence of connected reductive groups and Xss the open set of semistable points for the action of N on X then IHH*(Xss N) is (non-canonically) a direct summand of IHG*(Xss). The inclusion is provided by the decomposition theorem and certain resolutions of the action allow us to define projections.