Decomposition into irreducible of induced representations 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 ?
 A: Since it has not really been made completely explicit in the answers above, I point out that the specific question of whether your invariant subspace $W$ is irreducible or not is easier to check, at least in the case of finite groups. It rarely is. As is implicit in the earler answers, there is a formula due to Mackey for the norm (squared) of the induced character (and Mackey's formula works in more general contexts. In the case of finite groups, it is a double application of Frobenius reciprocity). We have $\langle {\rm Ind}_{H}^{G}(1), {\rm Ind}_{H}^{G}(1) \rangle = |T|$ , where $T$ is a full set of representatives for the distinct $(H,H)$-double cosets (so $G$ is the disjoint union $\cup_{t \in T} HtH ).$ The space $W$ is irreducible if and only if this norm is $2,$ which happens if and only if there are just two $(H,H)$-double cosets. This in turn happens precisely when $G$ is doubly transitive in the usual action on the ( say right) cosets of $H.$
A: You will have to compute the algebra of intertwiner
$$A = Endo_G( Ind_H^G 1) \cong \bigoplus\limits_{\pi} M_{m(\pi) \times m(\pi)}(\mathbb{C}),$$
where $Ind_H^G 1 = \bigoplus_\pi m(\pi) (V_\pi,\pi)$.
There are two thing, which I usually use: 


*

*There is a $*$ algebra isomorphism
$$ A \cong C[ H \backslash G /H]$$

*Use Frobenius reciprocity and the induction restriction formula
$$ A \cong Hom_H( 1, Res_H Ind_H^G 1) 
= \bigoplus_{\gamma \in H \backslash G /H} Hom_H ( 1, Ind_{H^g \cap H}^H 1).$$
For locally profinite groups, you do the same thing and everything what I said, remains true by replacing group rings by $C_c^\infty$ and isomorphism by dense embeddings.
