Polynomial functions for locally compact group actions

Abstract

Consider a locally compact group $G$ and a locally compact space $X$. A local right action of $G$ on $X$ is a continuous map $(x,p)\mapsto x\cdot p$ from an open subset $\Gamma$ of the Cartesian product $X\times G$ to $X$ satisfying certain obvious properties. A global right action of $G$ on $X$ gives rise to a global left action of $G$ on the space $C_c(X)$ of continuous complex functions with compact support in $X$ by the formula $p\,\cdot f:x\mapsto f(x\cdot p)$. In the case of a local action, one still can define $p\,\cdot f$ in $C_c(X)$ by this formula for $f\in C_c(X)$ and $p$ in a neighborhood $V_f$ of the identity in $G$. This yields a local left action of $G$ on $C_c(X)$. Given a local right action of $G$ on $X$, a function $f\in C_c(X)$ is called polynomial if there is a neighborhood $V$ of the identity, contained in $V_f$, and a finite-dimensional subspace $F$ of $C_c(X)$ containing all the functions $v\cdot f$ for $v\in V$. In this paper we study such polynomial functions. If $G$ acts on itself by multiplication, we are also interested in the local actions obtained by restricting it to an open subset of $G$. This is the typical situation that is encountered in our paper on bicrossproducts of groups with a compact open subgroup. In fact, the need for a better understanding of polynomial functions for that case has led us to develop the theory in general here.