A relation between invariant means on Lie groups and invariant means on their discrete subgroups

Abstract:

Let $G$ be a Lie group, and let $D$ be a discrete subgroup of $G$ such that the right coset space $D\backslash G$ has finite right-invariant volume. We will exhibit an injection of left-invariant means on ${l^\infty }(D)$ into left-invariant means on the left uniformly continuous bounded functions of $G$. When $G$ is an abelian Lie group with finitely many connected components, we also show surjectivity, and when $G$ is the additive group ${{\mathbf {R}}^n}$ and $D$ is ${{\mathbf {Z}}^n}$, the bijection will explicitly take the form of an integral over the unit cube ${[0,1]^n}$.