Given a symplectic vector space $(V,\omega)$ of dimension $2n$ ($V$ being a symplectic manifold), a smooth function $\psi \in C^\infty(M)$, a translation operator $T$ acting on $V$ via $(T(Y)\psi)(X) = \psi(X+Y)~e^{-\frac{i}{\hbar}\omega(Y,X)}$, $X,Y \in V$, where two translations combine as $$ T(Y)T(Z)=e^{\frac{i}{\hbar}\omega(X,Y)}T(Y+Z).$$
Let $T\subset \mathbb{C}$ denote the circle group, then the Heisenberg group may be given as $V \times T$, with group law $$(X,x)\cdot(Y,y) = (X+Y,xy~e^{\frac{i}{\hbar}\omega(X,Y)}).$$
I would like to show that the given translations form a reducible representation of the Heisenberg group, yet I'm not sure how to go about it. Naively, I assume the representation is $\rho: V\times T \rightarrow \mathcal{O} ~,~ (X,x) \mapsto xT(X)$. Now I have to find some non-trivial subspace of $V$, which is invariant under $\rho$. Does anyone have a clue for me? Thank you for your help.