Invariant Integration and Characters of Irreducible Representations

A fundamental tool in the study of compact groups (such as SU (2), tori and SO(n) for any n) is invariant integration. An integral on a group G allows us to dehne a complex vector space An integral invariant under multipli- 

However, a slight modification will bring the circle in line with the customary invariant integration. Parametrizing the circle by 

Let us double check that the integral is invariant under left multiplication. Any element of the group can be written for some to e R, so we have 

Note that the integral is invariant under right multiplication as well  

Right invariance follows from left invariance for all compact groups. The 

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