tailieunhanh - Equivariant structure constants for Hamiltonian-T-spaces

If there exists a set of canonical classes on a compact Hamiltonian-T -space in the sense of R Goldin and S Tolman, we derive some formulas for certain equivariant structure constants in terms of other equivariant structure constants and the values of canonical classes restricted to some fixed points. | Turkish Journal of Mathematics Research Article Turk J Math (2014) 38: 483 – 491 ¨ ITAK ˙ c TUB ⃝ doi: Equivariant structure constants for Hamiltonian-T -spaces Ho-Hon LEUNG∗ School of Liberal Arts and Sciences, Canadian University of Dubai, Dubai, United Arab Emirates Received: • Accepted: • Published Online: • Printed: Abstract: If there exists a set of canonical classes on a compact Hamiltonian- T -space in the sense of R Goldin and S Tolman, we derive some formulas for certain equivariant structure constants in terms of other equivariant structure constants and the values of canonical classes restricted to some fixed points. These formulas can be regarded as a generalization of Tymoczko’s results. Key words: Symplectic geometry, moment map, equivariant cohomology, equivariant structure constant 1. Introduction Let T be a compact torus with its Lie algebra t and lattice l ⊂ t. For a compact symplectic manifold (M, ω) equipped with a Hamiltonian-T -action, we have a moment map ϕ : M → t∗ , where t∗ is the dual of t. Then we have the following equation: ιXξ ω = −dϕξ , ∀ξ ∈ t, where Xξ denotes the vector field on M generated by the action and ϕξ : M → R is defined by ϕξ (x) = ⟨ϕ(x), ξ⟩. Here, ⟨., .⟩ is the natural pairing of t∗ and t. M is called a compact Hamiltonian-T -space. ϕξ is called the component of the moment map ϕ corresponding to the chosen element ξ ∈ t. Suppose that the component of the moment map is generic; that is, ⟨η, ξ⟩ ̸= 0 for each weight η ∈ l∗ ⊂ t∗ in the symplectic representation Tp M for every p in the T -fixed point set M T , and then ψ = ϕξ : M → R is a Morse function with the critical set M T . Under this situation, the Morse index of ψ at each p ∈ M T is even. Let λ(p) be half of the index of ψ at p . Let ∧− p be the product of all the individual weights of this representation. For each p ∈ M T , the natural inclusion map ip : p