Hyperbolic localization of intersection cohomology

For a normal variety X defined over an algebraically closed field with an action of the multiplicative group T = G_m, we consider the "hyperbolic localization" functor D^b(X) → D^b(X^T), which localizes using closed supports in the directions flowing into the fixed points, and compact supports in the directions flowing out. We show that the hyperbolic localization of the intersection cohomology sheaf is a direct sum of intersection cohomology sheaves.