Localization and a generalization of MacDonald's inner product

We find a limit formula for a generalization of MacDonald's inner product in finitely many variables, using equivariant localization on the Grassmannian variety, and the main lemma from \cite{Car}, which bounds the torus characters of the higher \c{C}ech cohomology groups. We show that the MacDonald inner product conjecture of type $A$ follows from a special case, and the Pieri rules section of MacDonald's book \cite{Mac}, making this limit suitable replacement for the norm squared of one, the usual normalizing constant.