Operational calculus for the Fourier transform on the group \(\mathrm{GL}(2,\mathbb{R})\) and the problem about the action of an overalgebra in the Plancherel decomposition (Q1755981)

Using the Plancherel theorem, the author relates the elements of the group \(\mathrm{GL}(2,\mathbb R)\) to representations in the space of functions on the real numbers \(\mathbb R\). The Fourier transformation based on this transforms differential operators on the space spanned by the group elements to differential-difference operators where the difference part is given by shift operators on the space of indices of the representation. This correspondence is formulated in the two central Theorems 1 and 2. In the following, the author deals with the overalgebra problem, trying to extend a spectral decomposition given on a subgroup to the whole Lie group. This is dealt with in different explicit examples of overalgebras of \(\mathrm{GL}(2,\mathbb R)\).