Symplectic structures on locally compact abelian groups and polarizations (Q1322373)

The goal of the note is the study of a family of locally compact abelian groups for which a nondegenerate symplectic structure admits a polarization. Let \(\mathcal L\) denote the class of locally compact Hausdorff, abelian and second countable groups. Let \({\mathcal L}_ 0\) be the class of groups \(X \in {\mathcal L}\) possessing the following properties: (i) The maximal compact connected subgroups of both \(X\) and its dual \(X^*\) are tori; (ii) \(\dim_{Q_ p} \text{hom}(X, Q_ p) < \infty\) for all primes \(p\); (iii) The subgroup \(\{x \in X : p.x = 0\}\) is finite for each prime \(p\). The main result of the present note is Theorem 1. If \(X \in {\mathcal L}_ 0\) and \(\omega\) is a nondegenerate symplectic structure on \(X\), then \((X,\omega)\) admits a polarization. Mention must be made of the fact that it is not known whether, for a general \(X \in {\mathcal L}\), a nondegenerate symplectic structure always admits a polarization.