The isotropy lattice of a lifted action
Miguel Rodriguez-Olmos
TL;DR
The paper addresses how to determine the isotropy lattice for lifted actions of a Lie group $G$ on $TM$ and $T^*M$ from the base action on $M$. It proves a precise criterion: $(L)\in I(G,TM)$ iff there exist $(H_1),(H_2)\in I(G,M)$ with $(H_1)\le (H_2)$ and $(K)\in I(H_2,\mathrm{ann}\mathfrak{h}_2)$ such that $L=H_1\cap K$, deriving this from a $G$-invariant metric and the Tube Theorem, and it provides an explicit algorithm to compute the lifted lattice. The framework extends to symplectic geometry via the cotangent lift and momentum map, introducing the $\mu$-lattice $I^\mu(G,M)$ and showing $I(G,\mathbf{J}^{-1}(0))=I(G,M)$ and $I(G,\mathbf{J}^{-1}(\mu))=I^\mu(G,M)$, with the isotropy of possible relative equilibria given by $\bigcup_{(H)\in I(G,M)} G\cdot I(H,\mathrm{ann}\,\mathfrak{h})$. These results inform singular reduction and constrain stabilizers and bifurcations in symmetric Hamiltonian systems.
Abstract
We obtain an algorithmic construction of the isotropy lattice for a lifted action of a Lie group $G$ on $TM$ and $T^*M$ based only on the knowledge of $G$ and its action on $M$. Some applications to symplectic geometry are also shown.