Harmonic Analysis on the Affine Group of the Plane

TL;DR

This paper explores the harmonic analysis of the affine group of the plane, providing a concrete realization of its unique square-integrable representation and decomposing the regular representation into these components.

Contribution

It explicitly constructs a concrete realization of the square-integrable representation of the affine group of the plane and decomposes the regular representation accordingly.

Findings

Explicit realization of the square-integrable representation on a specific Hilbert space

Decomposition of the regular representation into irreducible components

Identification of the unique square-integrable representation for the affine group

Abstract

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 $\sigma_2$.