Index problem for the metaplectic group. I.

We consider nonlocal elliptic operators in spaces of functions in $\mathbb{R}^n$, generated by differential operators and quantizations of linear canonical transformations. This theory includes as special cases, noncommutative tori of A.Connes, shift and dilation operators, etc. We explain index formulas, recently obtained in a joint work with Prof. E.Schrohe (Hannover).