This paper investigates ways to enlarge the Hamiltonian subgroup Ham of the symplectomorphism group Symp(M) of the symplectic manifold (M,ω) to a group that both intersects every connected component of Symp(M) and characterizes symplectic bundles with fiber M and closed connection form. As a consequence, it is shown that bundles with closed connection form are stable under appropriate small per...

In this paper we study symplectic maps with a continuous symmetry group arising by periodic forcing of symmetric Hamiltonian systems. By Noether’s Theorem, for each continuous symmetry the symplectic map has a conserved momentum. We study the persistence of relative periodic points of the symplectic map when momentum is varied and also treat subharmonic persistence and relative subharmonic bifu...

We interpret a result of Oehms as a statement about the symplectic ideal. We use this result to prove a double centraliser theorem for the symplectic group acting on ⊕s r=0 ⊗ V , where V is the natural module for the symplectic group. This result was obtained in characteristic zero by Weyl. Furthermore, we use this to extend to arbitrary connected reductive groups G with simply connected derive...

Let G be a compact connected semisimple Lie group. We extend the techniques of Weinstein [W] to give a construction in group cohomology of symplectic forms ω on ‘twisted’ moduli spaces of representations of the fundamental group π of a 2-manifold Σ (the smooth analogues of Hom(π1(Σ), G)/G) and on relative character varieties of fundamental groups of 2-manifolds. We extend this construction to e...

We study the nullcone for the symplectic group with respect to a joint action of the general linear group and the symplectic group. By extracting an algebra over a distributive lattice structure from the coordinate ring of the nullcone, we describe a toric degeneration and standard monomial theory of the nullcone in terms of double tableaux and integral points in a convex polyhedral cone.

There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a sym...

In this article, motivated by the study of symplectic structures on manifolds with boundary and systematic b $b$ -symplectic started in Guillemin, Miranda, Pires Adv. Math. 264 (2014), 864–896, we prove a slice theorem for Lie group actions manifolds.

To the best of our knowledge, the study of the local symplectic invariants of submanifolds of Euclidean space was initiated by Chern and Wang in 1947, [6]. They considered mainly the case of curves and hypersurfaces, and obtained structure equations defining a set of local symplectic differential invariants for these objects. We should explain at this stage that by “symplectic invariants” we me...

Abstract. The present paper discusses the question of formulating and solving minimal-distance problems over the group-manifold of real symplectic matrices. In order to tackle the related optimization problem, the real symplectic group is regarded as a pseudo-Riemannian manifold and a metric is chosen that affords the computation of geodesic arcs in closed forms. Then, the considered minimal-di...

Let n > 1 and G be the group SU(n) or Sp(n). This paper constructs compact symplectic manifolds whose symplectic quotient under a Hamiltonian G-action does not inherit the strong Lefschetz property .