Harmonic Analysis on the Affine Group of the Plane

For any natural number $n$, the group $G_n$ of all invertible affine transformations of $n$-dimensional Euclidean space has, up to equivalence, just one square-integrable representation and the left regular representation of $G_n$ is a multiple of this square-integrable representation. We provide a concrete realization $\sigma_2$ of this square-integrable representation of $G_2$ acting on the Hilbert space $L^2\big(\widehat{\mathbb{R}^2}\times\widehat{\mathbb{R}}\big)$. We explicitly decompose the Hilbert space $L^2(G_2)$ as a direct sum of left invariant closed subspaces on each of which the left regular representation acts as a representation equivalent to $\sigma_2$.