Induced representations for profinite groups 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?
 A: 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 finite-dimensional. This can be found under Peter-Weyl 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 Peter-Weyl or its finite-group analogue).
