  Let T be a maximal torus of a connected reductive group G that acts linearly
on a projective variety X so that all semi-stable points are stable. This paper
compares the integration on the geometric invariant theory quotient X//G of
Chow classes to the integration on the geometric invariant theory quotient X//T
of certain lifts of these classes twisted by the top Chern class of the T
-equivariant vector bundle induced by the quotient of the adjoint
representation on the Lie algebra of G by that of T . We provide a purely
algebraic proof that the ratio between any two such integrals is an invariant
of the group G and that it equals the order of the Weyl group whenever the root
system of G decomposes into irreducible root systems of type $A_n$, for any
natural numbers n. As a corollary, we are able to remove this restriction on
root systems by applying a related result of Martin from symplectic geometry.
