Zak transform associated with the Weyl transform and the system of twisted translates on R^{2n}

We introduce the Zak transform on $L^2({\mathbb{R}}^{2n})$ associated with the Weyl transform. By making use of this transform, we define a bracket map and prove that the system of twisted translates $\{T_{(k,l)}^t\phi :k,l\in {\mathbb{Z}}^n\}$ is a frame sequence iff $0<A\le [\phi ,\phi ](\xi ,{\xi }^{{}^{\mathrm{\prime }}})\le B<\mathrm{\infty },$ for a.e $(\xi ,{\xi }^{{}^{\mathrm{\prime }}})\in {\mathrm{\Omega }}_{\phi },$ where ${\mathrm{\Omega }}_{\phi }=\{(\xi ,{\xi }^{{}^{\mathrm{\prime }}})\in {\mathbb{T}}^n\times {\mathbb{T}}^n:[\phi ,\phi ](\xi ,{\xi }^{{}^{\mathrm{\prime }}})\mathrm{\ne }0\}$. We also prove a similar result for the system $\{T_{(k,l)}^t\phi :k,l\in {\mathbb{Z}}^n\}$ to be a Riesz sequence. For a given function belonging to the principal twisted shift-invariant space $V^t(\phi )$, we find a necessary and sufficient condition for the existence of a canonical biorthogonal function. Further, we obtain a characterization for the system $\{T_{(k,l)}^t\phi :k,l\in \mathbb{Z}\}$ to be a Schauder basis for $V^t(\phi )$ in terms of a Muckenhoupt ${\mathcal{A}}_2$ weight function.