Banach Convolution Modules of Group Algebras on Covariant Functions of Characters of Normal Subgroups

This paper investigates structure of Banach convolution modules induced by group algebras on covariant functions of characters of closed normal subgroups. Let $G$ be a locally compact group with the group algebra $L^1(G)$ and $N$ be a closed normal subgroup of $G$. Suppose that $\xi:N\to\mathbb{T}$ is a continuous character, $1\le p<\infty$ and $L_\xi^p(G,N)$ is the $L^p$-space of all covariant functions of $\xi$ on $G$. It is shown that $L^p_\xi(G,N)$ is a Banach $L^1(G)$-module. We then study convolution module actions of group algebras on covariant functions of characters for the case of canonical normal subgroups in semi-direct product groups.