Flexibility of the Hamiltonian adjoint action and classification of bi-invariant metrics
On an open, connected symplectic manifold $(M,\omega)$, the group of Hamiltonian diffeomorphisms forms an infinite-dimensional Fr\'echet Lie group with Lie algebra $C^{\infty}_c(M)$ and adjoint action given by pullbacks. We prove that this action is flexible: for any non-constant $u \in C^{\infty}(M)$, every $f \in C^{\infty}_c(M)$ can be expressed as a weighted finite sum of elements from the adjoint orbit of $u$, with total weight bounded by constant multiple of $\|f\|_{\infty} + \|f\|_{L^1}$. Consequently, all $\mathrm{Ham}(M,\omega)$-invariant norms on $C^{\infty}_c(M)$ are dominated by a sum of $L^{\infty}$ and $L^1$ norms. As an application, we classify up to equivalence all bi-invariant pseudo-metrics on the group of Hamiltonian diffeomorphisms of an exact symplectic manifold, answering a question of Eliashberg and Polterovich.