Left regular representation

Summary

The left regular representation of a locally compact group is the action of the group on square-integrable functions on itself by left translation. It is a unitary representation, and it contains, in a suitable sense, the irreducible representations that matter for harmonic analysis on the group. Its decomposition into irreducibles is the content of Plancherel theorems. Many properties of the group, such as amenability, can be read off from it.