Harmonic Analysis on the Affine Group of the Plane

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 $$\sigma $$ of this distinguished representation in the two-dimensional case. 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 $$\sigma $$ .