$G$-isotropy of $T$-relative equilibria within manifolds tangent to spaces with linearly independent weights

Abstract

We investigate the generic local structure of relative equilibria in Hamiltonian systems with symmetry $G$ near a completely symmetric equilibrium, where $G$ is compact and connected. Fix a maximal torus $T \subset G$ and identify the equilibrium with the origin within a symplectic representation of $G$. By a previous result, generically, for each $\xi \in \mathfrak t$ such that $V_0:= \ker \mathrm d^2(h-\mathbf J^\xi)(0)$ has linearly independent weights, there is a manifold tangent to $V_0$ that consists of relative equilibria with generators in $\mathfrak t$. Here we determine their isotropy with respect to $G$. The main result asserts that for each of these manifolds of $T$-relative equilibria, there is a local diffeomorphism to its tangent space at $0$ that preserves the isotropy groups. We will then deduce that the $G$-orbit of the union of these manifolds is stratified by isotropy type. The stratum given by the relative equilibria of type $(H)$ has the dimension $\dim G -\dim H + \dim (\mathfrak t')^L$, where $\mathfrak t' \subset \mathfrak t$ is the orthogonal complement of $\mathfrak h \cap \mathfrak t$ and $L$ is the minimal adjoint isotropy subgroup of an element of $\mathfrak t$ with $H \subset L$. In the end, we consider some examples of these manifolds of $T$-relative equilibria that contain points with the same isotropy type with respect to the $T$-action but different isotropy type with respect to $G$.