For an actual maximal simple character \(\theta\) of \({\mathrm{GL}}_n(K)\), classical simple-type theory supplies
\[
H_\theta^1\subseteq J_\theta^1\subseteq J_\theta,
\qquad V_\theta=J_\theta^1/H_\theta^1.
\]
The commutator pairing is encoded by
\[
\psi_0(h_\theta(\bar x,\bar y))=\theta([x,y]).
\]
It is nondegenerate and alternating. The corresponding irreducible Heisenberg representation \(\eta_\theta\) of \(J_\theta^1\) contains \(\theta\) and satisfies
\[
\dim\eta_\theta=\sqrt{|J_\theta^1/H_\theta^1|}.
\]
The canonical beta extension supplies a normalized extension of this representation using the declared type-theory data and finite corrections.
Scope
An arbitrary abstract symplectic space is not automatically a quotient of an actual simple character. An equivalent inducing carrier need not have this original quotient or a newly invented canonical beta extension.