A ratio of integration between quotients in geometric invariant theory

Zachary Maddock

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arXiv · 1210.4253

A ratio of integration between quotients in geometric invariant theory

Zachary Maddock

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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.

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Zachary Maddock. 2014-01-16. A ratio of integration between quotients in geometric invariant theory. https://arxiv.org/abs/1210.4253

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