Symmetry of a symplectic toric manifold

Abstract

The action of a torus group $T$ on a symplectic toric manifold $(M,\omega)$ often extends to an effective action of a (non-abelian) compact Lie group $G$. We may think of $T$ and $G$ as compact Lie subgroups of the symplectomorphism group $Symp(M,\omega)$ of $(M,\omega)$. On the other hand, $(M,\omega)$ is determined by the associated moment polytope $P$ by the result of Delzant. Therefore, the group $G$ should be estimated in terms of $P$ or we may say that a maximal compact Lie subgroup of $Symp(M,\omega)$ containing the torus $T$ should be described in terms of $P$. In this paper, we introduce a root system $R(P)$ associated to $P$ and prove that any irreducible subsystem of $R(P)$ is of type A and the root system $\Delta(G)$ of the group $G$ is a subsystem of $R(P)$ (so that $R(P)$ gives an upper bound for the identity component of $G$ and any irreducible factor of $\Delta(G)$ is of type A). We also introduce a homomorphism from the normalizer of $T$ in $G$ to an automorphism group $Aut(P)$ of $P$, which detects the connected components of $G$. Finally we find a maximal compact Lie subgroup $G_{\max}$ of $Symp(M,\omega)$ containing the torus $T$.