Coisotropic subspace

The following conditions are equivalent: $W$ is coisotropic; $W^\omega\subseteq W$; the symplectic orthogonal $W^\omega$ is isotropic; and $\ker(\omega|_W)$ has dimension equal to $\operatorname{codim}W$. In particular, a coisotropic subspace of a $2n$-dimensional symplectic vector space has dimension at least $n$.

The whole space $V$ is coisotropic because $V^\omega=\{0\}$. Every hyperplane in a finite-dimensional symplectic vector space is coisotropic. A Lagrangian subspace is both isotropic and coisotropic, since $W=W^\omega$. By contrast, a proper symplectic subspace is not coisotropic: nondegeneracy gives $W\cap W^\omega=\{0\}$, so $W^\omega\subseteq W$ would force $W=V$.

For $u,v\in W$, define $\bar\omega([u],[v])=\omega(u,v)$ on $W/W^\omega$. This is well defined because changing either representative by an element of $W^\omega$ leaves the value unchanged. Its kernel is zero by construction. The same mechanism globalizes, under regularity hypotheses, to the reduction of a coisotropic submanifold by its characteristic foliation.

Some sources write “co-isotropic.” The definition here is purely linear and finite-dimensional. For a coisotropic submanifold $C$ of a symplectic manifold, the condition is imposed pointwise as $(T_pC)^\omega\subseteq T_pC$, and forming a smooth quotient requires additional global hypotheses. The linear theory is developed in Cannas da Silva, §1.1.

1. Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: §1.1, isotropic, coisotropic, and Lagrangian subspaces.
2. Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. DOI record. Relevant: linear symplectic algebra and reduction.