Localization and a generalization of MacDonald's inner product

TL;DR

This paper derives a limit formula for a generalized MacDonald's inner product using equivariant localization, linking it to MacDonald's conjecture and Pieri rules, providing a new perspective on symmetric function inner products.

Contribution

It introduces a limit formula for a generalized MacDonald's inner product via equivariant localization, connecting it to MacDonald's conjecture and Pieri rules.

Findings

Established a limit formula for the generalized inner product.

Connected the formula to MacDonald's conjecture of type A.

Linked the Pieri rules to the limit formula.

Abstract

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.