Singular Cotangent Bundle Reduction and Polar Actions
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Singular Cotangent Bundle Reduction and Polar Actions Xiaoyang Chen1 · Jianyu Ou2 Received: 14 December 2017 © Mathematica Josephina, Inc. 2019
Abstract A conjecture of Lerman, Montgomery and Sjamaar states that two singular symplectic reductions T ∗ M G and T ∗ N H are isomorphic if M/G is diffeomorphic to N /H as stratified spaces. We confirm this conjecture under the assumptions that the action G × M → M is polar with a section N and generalized Weyl group H . Keywords Polar action · Singular symplectic reduction · Chevalley Restriction Theorem Mathematics Subject Classification 53C20 · 53D20
1 Introduction Let G be a compact Lie group acting isometrically on a complete Riemannian manifold (M, g). It is well known that the lifting action on the cotangent bundle T ∗ M with its canonical symplectic structure ω is a Hamiltonian action with a moment map given by u : T ∗ M → g∗ with u X (x, ξ ) = ξ, X ∗ (x),
(1.1)
where g is the Lie algebra of G, g∗ is the dual of g and u X (x, ξ ) = u(x, ξ ), X , X ∈ g. Moreover, X ∗ is the vector field on M generated by X . The moment map satisfies the following equations: du X = i X # ω,
B
Jianyu Ou [email protected] Xiaoyang Chen [email protected]
1
School of Mathematical Sciences, Institute for Advanced Study, Tongji University, Shanghai, China
2
Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China
123
X. Chen, J. Ou
u(g · (x, ξ )) = Adg∗ · u(x, ξ ), ∀g ∈ G, where X # is the vector field on T ∗ M generated by X . The symplectic reduction T ∗ M G := u −1 (0)/G is not a smooth manifold in general. However, it is a stratified symplectic space defined in [20]. The reader is referred to [20] for the precise definition of stratified symplectic spaces. Singular symplectic reductions have played an important role in geometric quantization [10]. Following [20], we define a function f : T ∗ M G → R to be smooth if there exists a function F ∈ C ∞ (T ∗ M)G with F|μ−1 (0) = π ∗ f , where π : μ−1 (0) → μ−1 (0)/G is the projection map. In other words, C ∞ (T ∗ M G) is isomorphic to C ∞ (T ∗ M)G /I G , where I G is the ideal of G-invariant smooth functions on T ∗ M vanishing on μ−1 (0). The algebra C ∞ (T ∗ M G) inherits a Poisson algebra structure from C ∞ (T ∗ M). Let G and H be Lie groups and M, resp. N , be smooth manifolds on which G, resp. H act properly. The stratified symplectic spaces T ∗ M G and T ∗ N H are isomorphic if there exists a homeomorphism φ : T ∗ M G → T ∗ N H and the pullback map φ ∗ : C ∞ (T ∗ N H ) → C ∞ (T ∗ M G), f → f ◦ φ is an isomorphism of Poisson algebras. In [9, p. 13, Conjecture 3.7], they made the following conjecture. Conjecture 1.1 Let G and H be Lie groups and M, resp. N be smooth manifolds on which G, resp. H act properly. Assume that the orbit spaces M/G and N /H are diffeomorphic in the sense that there exists a homeomorphism φ : M/G → N /H such that the pullback map φ ∗ is an isomorphism from C ∞ (N /H ) := C ∞ (N ) H to C ∞ (M/G) := C ∞ (M)G . Then T ∗ M G and T ∗ N H are is
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