ar X iv : 0 90 6 . 43 97 v 1 [ m at h . G R ] 2 4 Ju n 20 09 Locally Compact Abelian Groups with symplectic self - duality

Is every locally compact abelian group which admits a symplectic self-duality isomorphic to the product of a locally compact abelian group and its Pontryagin dual? Several sufficient conditions, covering all the typical applications are found. Counterexamples are produced by studying a seemingly unrelated question about the structure of maximal isotropic subgroups of finite abelian groups with symplectic self-duality (where the original question always has an affirmative answer). 1. The main questions Let T denote the circle group, thought of as R/Z. The Pontryagin dual of a locally compact abelian group is the locally compact abelian group L̂ of continuous homomorphisms L → T, endowed with the compact open topology. Definition (Symplectic self-duality). A self-duality of a locally compact abelian group L is an isomorphism ∇ : L → L̂. If ∇(x)(x) = 0 for all x ∈ L, then ∇ is called a symplectic self-duality. Definition (Isotropic subgroup). Let ∇ be a self-duality of a locally compact abelian group L. A subgroup N of L is said to be isotropic if ∇(x)(y) = 0 for all x, y ∈ N . Definition (Isomorphism of self-dualities). Let ∇ and ∇ be self-dualities of locally compact abelian groups L and L respectively. The pairs (L,∇) and (L,∇) are said to be isomorphic if there exists an isomorphism φ : L → L such that ∇(φ(x))(φ(y)) = ∇(x)(y). If M and M ′ are subgroups in L and L respectively, and φ above can be chosen so that φ(M) = M , then (L,∇,M) and (L,∇,M ) are said to be isomorphic triples. Example 1.1. For any locally compact abelian group A, let L = A× Â, and define ∇ : L → L̂ by ∇(x, χ)(y, λ) = χ(y)− λ(x) for x, y ∈ A and χ, λ ∈ Â. 2000 Mathematics Subject Classification. 22B05.