I am studying the representation theory for a compact topological group. I have already studied the representation theory for fininte groups. There we study the module C(G) for the finite group G. But I can't understand why are we studying L_2(G) instead of L_1(G) or L_p(G) for some other p?
Any help would be appreciated. Thanks in advance.
There's no reason you can't study $L^p(G)$ for other values of $p$, and some people do. The reasons to focus on $L^2(G)$ have nothing to do with the fact that $G$ is a group: for arbitrary measure spaces $X$, people tend to be more interested in $L^2(X)$ than $L^p(X)$ for other values of $p$. The reason is that $L^2(X)$ is a Hilbert space (whereas $L^p(X)$ for general $p$ is just a Banach space), which gives you lots of extra structure to play with (for instance, you can talk about things like orthonormal bases, and adjoint operators that are still defined on the same space).