The Fourier Transform on the Group mathrm {GL}_2({mathbb R }) and the Action of the Overalgebra mathfrak {gl}_4

Abstract

We define a kind of ’operational calculus’ for the Fourier transform on the group mathrm {GL}_2({mathbb R }). Namely, mathrm {GL}_2({mathbb R }) can be regarded as an open dense chart in the Grassmannian of 2-dimensional subspaces in {mathbb R }^4. Therefore the group mathrm {GL}_4({mathbb R }) acts in L^2 on mathrm {GL}_2({mathbb R }). We transfer the corresponding action of the Lie algebra mathfrak {gl}_4 to the Plancherel decomposition of mathrm {GL}_2({mathbb R }), the algebra acts by differential-difference operators with shifts in an imaginary direction. We also write similar formulas for the action of mathfrak {gl}_4oplus mathfrak {gl}_4 in the Plancherel decomposition of mathrm {GL}_2({mathbb C }).