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 $\mathfrak{k}$. Given a Hamiltonian action of $K$ on a compact symplectic manifold $X$ with moment map $\mu : X \to \mathfrak{k}^\ast$, the normsquare ${\lVert \mu \rVert}^2$ of $\mu$ defines a Morse stratification $\lbrace S_\beta : \beta \in \mathcal{B} \rbrace$ of $X$ by locally closed symplectic submanifolds of $X$ such that the stratum to which any $x \in X$ belongs is determined by the limiting behaviour of its downwards trajectory under the gradient flow of ${\lVert \mu \rVert}^2$ with respect to a suitably compatible Riemannian metric on $X$. The open stratum $S_0$ retracts $K$-equivariantly via this gradient flow to the minimum $\mu^{-1} (0)$ of ${\lVert \mu \rVert}^2$ (if this is not empty). If $\beta \neq 0$ the usual ‘symplectic quotient’ $( S_\beta \cap \mu^{-1} (0)) / K$ for the action of $K$ on the stratum $S_\beta$ 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 $\beta \neq 0$. There is an analogous infinite-dimensional picture for the Yang–Mills functional over a Riemann surface with strata determined by Harder–Narasimhan type.

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Received 14 December 2017

Accepted 21 May 2018

Published 1 October 2020