Abelian instances of nonabelian symplectic reduction

Abstract

Let $\mathbb{G}$ be a Lie group with a normal abelian subgroup $\mathbb{A}$, and let $(M,\omega)$ be a symplectic manifold endowed with a Hamiltonian $\mathbb{G}$-action. We investigate conditions under which symplectic reduction by $\mathbb{G}$ coincides with the symplectic reduction by the abelian subgroup $\mathbb{A}$. Using the reduction-by-stages framework (Marsden et al Springer Notes in Math., 1913, (2007)), we prove that, under a mild assumption, the corresponding reduced spaces are symplectomorphic if and only if they have the same dimension. Both this assumption and the dimension condition depend only on the groups $\mathbb{G}$ and $\mathbb{A}$, and on the momentum value $\mu\in \mathfrak{g}^*$ at which the symplectic reduction by $\mathbb{G}$ is performed; in particular, they are independent of the symplectic manifold $(M,\omega)$. We then provide a broad class of examples by identifying a large family of nilpotent Lie groups, including classical Carnot groups such as the Heisenberg group and jet-space $\mathcal{J}^k(\mathbb{R}^n,\mathbb{R}^m)$, for which the two reduced spaces are symplectomorphic for generic momentum values.