We obtain the symplectic group as an amalgam of low rank subgroups akin to Levi components. We do this by having the group act flag-transitively on a new type of geometry and applying Tits’ lemma. This provides a new way of recognizing the symplectic groups from a small collection of small subgroups.

Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, we prove a product formula for the Conley–Zehnder index of continuous paths with arbitrary endpoints in the symplectic group. We discuss two applications of the formula, to the metaplectic group and to periodic solutions of Hamiltonian systems.

Let T ⊂ T S be the bundle of unit discs, which is a compact symplectic manifold with boundary. Let S be the group of symplectic automorphisms of T which are trivial near ∂T . In [5] it was proved that π0(S) contains an element of infinite order. This element is given by the ‘generalized Dehn twist’ τ which is the monodromy map of a quadratic singularity [2] (see also [4]). The fact that [τ ] ha...

If π is a finite symplectic translation plane, it is shown that any affine homology group is cyclic and has order dividing the order of the kernel homology group. This criterion provides a means to ensure that a given spread is not symplectic. Furthermore, a variety of symplectically paired André spreads are constructed.

We determine an explicit formula for a Rankin-Cohen bracket for Siegel modular forms of degree n on a certain subgroup of the symplectic group. Moreover, we lift that bracket via a Poincaré series to a Siegel cusp form on the full symplectic group.

We generalize the concept of global moment maps to local moment maps whose different branches are labelled by the elements of the fundamental group of the underlying symplectic manifold. These branches can be smoothly glued together by employing fundamental-group-valued C̆ech cocycles on the phase space. In the course of this work we prove a couple of theorems on the liftability of group actions...

In this note we give a structure theorem for a finite-dimensional subgroup of the automorphism group of a compact symplectic manifold. An application of this result is a simpler and more transparent proof of the classification of compact homogeneous spaces with invariant symplectic structures. We also give another proof of the classification from the general theory of compact homogeneous spaces...

We prove several multiplicity one theorems in this paper. For k a local field not of characteristic two, and V a symplectic space over k, any irreducible admissible representation of the symplectic similitude group GSp(V ) decomposes with multiplicity one when restricted to the symplectic group Sp(V ). We prove the analogous result for GO(V ) and O(V ), where V is an orthogonal space over k. Wh...

Let G be a complex symplectic group. We construct a certain Gvariety N, which shares many nice properties with the nilpotent cone of G. For example, it is normal, has finitely many G-orbits, and admits a G-equivariant resolution of singularity in a similar way to the Springer resolution. Our new G-variety admits an action of G× C × C instead of the G×C-action on the nilpotent cone. This enables...

The automorphism group of a locally conformal symplectic structure is studied. It is shown that this group possesses essential features of the symplec-tomorphism group. By using a special type of cohomology the ux and Calabi homomorphisms are introduced. The main theorem states that the kernels of these homomorphisms are simple groups, for the precise statement see chapter 7. Some of the method...