If we consider an induced representation $Ind_{H}^{G}1_{H}$ of finite groups, where $1_{H}$ is the trivial caracter of $H$, how we decompose this representation into irreducible ? There is a decomposition $Ind_{H}^{G}1_{H}=V\oplus W$, where $V$ is formed of canstant functions on $G$ and $W$ is formed of functions $f:G\longrightarrow\mathbb{C}$, $f\in Ind_{H}^{G}1_{H}$, such taht $$\sum_{x\in G/H}f(x)=0$$ but I don't know if $W$ is irreducible and if it is not irreducible how decomposed there into irreducible ?
The question arises also for profinite groups : If $G$ is totaly disconnected compact group and $H$ is an open subgroup, how to decompose the representation $Ind_{H}^{G}1_{H}$ into irreducibles ?