  We compute explicitly the equivariant Hirzebruch $\chi_y$-characteristic of
an equivariant complex line bundle over a toric manifold and state a weighted
version of the quantization commutes with reduction principle in symplectic
geometry. Then, we give a weighted decomposition formula for any simple
polytope in $\R^n$. This formula generalizes a polytope decomposition due to
Lawrence [10] and Varchenko [14] and extends a previous weighted version
obtained by Karshon, Sternberg and Weitsman [9].
