A ratio of integration between quotients in geometric invariant theory

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.

This final version, which will soon appear in Transformation Groups, involves a thorough rewrite of the previous version, and includes a major simplification of Section 4. The title of this arXiv post has been changed from the original working title to match the title to appear in the published version

References in corpus (3)

• Two Dimensional Gauge Theories Revisited
• Quotients by Groupoids
• Symplectic quotients by a nonabelian group and by its maximal torus