A ratio of integration between quotients in geometric invariant theory

TL;DR

This paper establishes an algebraic relationship between integrals over GIT quotients with respect to a reductive group G and its maximal torus T, revealing invariance properties and explicit ratios linked to Weyl group order, extending previous geometric results.

Contribution

It provides a purely algebraic proof that the ratio of certain integrals on GIT quotients is a G-invariant and equals the Weyl group order for specific root systems, generalizing prior geometric findings.

Findings

The ratio of integrals is G-invariant.

The ratio equals the Weyl group order for type A root systems.

The result extends to all root systems using symplectic geometry techniques.

Abstract

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.