Symplectic quotients of unstable Morse strata for normsquares of moment maps

Let K be a compact Lie group and fix an invariant inner product on its Lie algebra k. Given a Hamiltonian action of K on a compact symplectic manifold X with moment map µ: X → k , the normsquare ||µ|| of µ defines a Morse stratification {S : β ∈ B} of X by locally closed symplectic submanifolds of X such that the stratum to which any x ∈ X belongs is determined by the limiting behaviour of its downwards trajectory under the gradient flow of ||µ|| with respect to a suitably compatible Riemannian metric on X. The open stratum S retracts K-equivariantly via this gradient flow to the minimum µ (0) of ||µ|| (if this is not empty). If β ≠ 0 the usual ‘symplectic quotient’ (S ∩ µ (0))/K for the action of K on the stratum S is empty. Nonetheless, motivated by recent results in non-reductive geometric invariant theory, we find that the symplectic quotient construction can be modified to provide natural ‘symplectic quotients’ for the unstable strata with β ≠ 0. There is an analogous infinite-dimensional picture for the Yang–Mills functional over a Riemann surface with strata determined by Harder–Narasimhan type. ∗ 2 2 −1 2 −1 β 0 β β