Let (V,ω) be a symplectic vector space. The Heisenberg group H(V) associated to V is a nilpotent real Lie group whose underlying manifold is V×R and a group law given by

The Lie algebra can be canonically identified with the manifold, so to distinguish between elements of the Lie algebra and the group, use exponential notation.

An admissible complex linear structure on V is a linear complex structure J such that

and

Denote by V_J the complex vector space induced by J. Then we have a Hermitian form.