Regular Representation of infinite groups

Question

I am aware of the importance of the regular representation of finite groups, but I am curios if there is an analogous notion for infinite groups. How is the regular representation of a infinite group defined? Does it decompose as the sum of all irreducible representations? Can we generalise this to a regular module of an algebra?

Answer 1

For an abstract infinite group $G$ the definition is the same as in the finite case; over a field $k$ you can always consider the action of the group algebra $k[G]$ on itself by left multiplication (and this generalizes to any ring whatsoever). It is never completely reducible if $G$ is infinite; see this math.SE answer. Every irreducible representation is (cyclic, hence) a quotient of $k[G]$ but the surjection $k[G] \to V$ need not split; it never splits, for example, if $G = \mathbb{Z}$.

In a positive direction, if $G$ is a compact (Hausdorff) topological group it has a regular representation on $L^2(G)$ where $G$ is equipped with Haar measure. By the Peter-Weyl theorem this representation decomposes as a (Hilbert space) direct sum of all the irreducible representations of $G$, and as in the finite group case a given representation $V_i$ occurs with multiplicity $\dim V_i$. For example when $G = S^1$ this recovers the (bare bones of the) theory of Fourier series.