I was wondering if there is any application for induced representations of profinite groups, for example in Galois theory or number theory? Is it a good research idea? Do you know any paper discussing this problem or related problem or at least unitary representations of profinite groups?
I claim that this is not more difficult/general then studying the induced representation of a finite group. (Edit: I am talking about complex representations only.) Every (continuous) representation of a compact group is unitarizable, hence it is sufficient to study the unitary representations in this case. They also decompose into irreducible ones. The irreducible ones are finitedimensional. This can be found under PeterWeyl theorem. For a profinite group $G$, an irreducible representation $\sigma$ has moreover an open kernel, i.e., the kernel is a finite index, normal subgroup. By Frobenius reciprocity, $\sigma$ is contained in $Ind_{kern( \sigma)}^G 1$. Because $kern( \sigma)$ is a finite index, normal subgroup, the representation $Ind_{kern( \sigma)}^G 1$ decomposes in the same way as does the right regular representation of $H=kern( \sigma) \backslash G$, i.e., every irreducible representation of $H$ occurs with multiplicity equals the dimension (again PeterWeyl or its finitegroup analogue). 

