My major is physics. I need to use some tools in group theory, but I am really confused by the trace in compact infinite groups. The following is my question:
In discrete group theory, the irreducible representation of identity group element $e$ is always an identity matrix. So the trace of $e$ under regular representation is the order of the group : $\chi(e) = |G|$. I hope to get similar result for Lie group. For example, SO(2) has infinite number of irreducible representations, $D^{(m)}$, where $m=0,\pm 1, \pm 2,\cdots $. All of them are 1-dimensional $D^{(m)} = e^{im\phi}$. Here $\phi$ is the rotation angle. For element $R(\phi)$ in SO(2), the trace in $m$th irreducible representation should be $\chi(R(\phi))=e^{im\phi}$, if we sum all these traces up, we get the trace in regular representation. So what is the trace of $e$ in the regular representation ? It seems to be infinite.