T-Weyl calculus

Let $\left( W,\sigma \right)$ be a symplectic vector space and let $$ be a linear map that satisfies a certain condition of non-degeneracy. We define the Schur multiplier $\omega _{\sigma ,T}$ on $W$. To this multiplier we associate a $\omega _{\sigma ,T}$-representation and and we build the $T$-Weyl calculus, $\mathrm{Op}_{\sigma ,T}$, whose properties are are systematically studied further.