Images of skyscraper sheaves on toric resolutions: cohomology distribution

Let G be a finite abelian subgroup  of SL_3(C). A famous theorem of Bridgeland, King and Reid (BKR) establishes a derived equivalence between triangulated categories D^b_G(Coh(C^3)) and D^b(Coh(Y)) (here Y is the G-Hilbert scheme). Any nontrivial irreducible representation of G gives rise to a skyscraper sheaf supported at 0 (equivariant coherent sheaf on C^3). It is known that the image of this sheaf under BKR equivalence is a pure sheaf supported in degree 0 or -1.  It is natural to study this dichotomy more scrupulously and that is exactly what we are going to do.