The geometry of special symplectic representations

TL;DR

This paper explores a class of symplectic Lie algebra representations that exhibit exceptional algebraic and geometric properties, including unique decompositions and special geometric structures like pseudo-Kähler metrics.

Contribution

It introduces a new class of symplectic representations with properties analogous to classical forms and provides explicit formulas for their decomposition and geometric structures.

Findings

All nonzero orbits are coisotropic.

Covariants satisfy generalized classical identities.

Existence of special geometric structures such as pseudo-Kähler metrics.

Abstract

We show there is a class of symplectic Lie algebra representations over any field of characteristic not 2 or 3 that have many of the exceptional algebraic and geometric properties of both symmetric three forms in two dimensions and alternating three forms in six dimensions. All nonzero orbits are coisotropic and the covariants satisfy relations generalising classical identities of Eisenstein and Mathews. The main algebraic result is that suitably generic elements of these representation spaces can be uniquely written as the sum of two elements of a naturally defined Lagrangian subvariety. We give universal explicit formulae for the summands and show how they lead to the existence of geometric structure on appropriate subsets of the representation space. Over the reals this structure reduces to either a conic, special pseudo-K\" ahler metric or a conic, special para-K\" ahler metric.