Symmetry of a Symplectic Toric Manifold

The action of a torus group T on a symplectic toric manifold (M, ω) often extends to an effective action of a (nonabelian) compact Lie group G. We may think of T and G as compact Lie subgroups of the symplectomorphism group Symp(M, ω) of (M, ω). On the other hand, (M, ω) is determined by the associated moment polytope P by the result of Delzant [4]. Therefore, the group G should be estimated in terms of P or we may say that a maximal compact Lie subgroup of Symp(M, ω) 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 ∆(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 ∆(G) is of type A). We also introduce a homomorphism D from the normalizer NG(T ) 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 Gmax of Symp(M, ω) containing the torus T .