Operational Calculus for the Fourier Transform on the Group $\operatorname{GL}(2,\mathbb{R})$ and the Problem about the Action of an Overalgebra in the Plancherel Decomposition

The Fourier transform on the group $\operatorname{GL}(2,\mathbb{R})$ of real $2\times2$ matrices is considered. It is shown that the Fourier images of polynomial differential operators on $\operatorname{GL}(2,\mathbb{R})$ are differential-difference operators with coefficients meromorphic in the parameters of representations. Expressions for operators contain shifts in the imaginary direction with respect to the integration contour in the Plancherel formula. Explicit formulas for the images of partial derivations and multiplications by coordinates are presented.