Journal of the
Korean Mathematical Society JKMS

This paper presents a systematic study for theoretical aspects of a unified approach to the abstract relative Fourier transforms over canonical homogeneous spaces of semi-direct product groups with Abelian normal factor. Let $H$ be a locally compact group, $K$ be a locally compact Abelian (LCA) group, and $\theta:H\to {\rm Aut}(K)$ be a continuous homomorphism. Let $G_\theta=H\ltimes_\theta K$ be the semi-direct product of $H$ and $K$ with respect to $\theta$ and $G_\theta/H$ be the canonical homogeneous space (left coset space) of $G_\theta$. We introduce the notions of relative dual homogeneous space and also abstract relative Fourier transform over $G_\theta/H$. Then we study theoretical properties of this approach.

Keywords: canonical homogeneous space, dual homogeneous space, relative convolution, relative Fourier transform, Plancherel formula, semi-direct product groups

MSC numbers: Primary 43A85; Secondary 43A15