2012/10/16 by Zachary Maddock, Maddock, Zachary · 1 citation
Mathematics · #14C15 (Secondary) #14L24 (Primary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1210.4253
openalex publication_date 2012/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 An, 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.