Suppose $K=\mathbb{Q}\_p$ ist the p-adic field and $G$ a finite group with a normal subgroup $S$. I would like to get all the irreducible representations of $G$ over $K$ .


I tried to solve this by using induction from the irreducible representations of $S$ and then checked with Mackey if they are irreducible or not.


What can one do if the induced representations decomposes into two irreducibles? Is there some theorem which tells, how many of them are different and how hey decomposes?



Representations over $\mathbb{Q}_p$
For finite groups and a field $F$ of characteristic zero, you can identify the $G$-endomorphisms
of two induced representations
$$ Hom\_G( I\_H^G \pi, I\_K^G \sigma) $$
with the space of functions $f: H \backslash G / K \rightarrow Hom\_F(V\_\pi, V\_\sigma\*)$ with $f(hgk)= \pi(h) f(g) \sigma(k)$.
If $H=G$ and $\pi = \sigma$, i.e., the case you are interested in, everything becomes a convolution algebra. If you find a basis, they give you projections to irreducible components, and you can decompose explicitly.


Equivalently, you could use Frobenius reciprocity
$$ Hom\_H( \pi, R\_H I\_K^G \sigma) $$
with Mackey induction restriction formula giving you a decomposition
$$R\_H I\_K^G \sigma$$
in terms of $H\backslash G/K$.


All this is very combinatorial. There is no general algorithm for general $G$ even in the case of complex representation, e.g., nobody knows how to decompose the parabolic induction in $GL\_3(\mathbb{Z} / p^N)$. So, one might say decomposing induced representations into irreducibles is an art rather then a theory.


The situation is slightly different in the special case, where $H$ is **normal**. Then induction exhaust all irreducible representations and it is easy to parametrize them. However, also here you encounter the same difficulties only later:( To understand this special case, I suggest you should learn Clifford theory.


*Edit in response to a comment:* Let $N$ be a normal subgroup. $G$ acts on the representation $\pi$ of $N$ via $g: \pi \mapsto \pi^g(x) = \pi(gxg^{-1})$.



> 
> 1. $Ind\_N^G \pi$ decomposes with single multiplicity iff $End\_G( I\_N^G \pi)$ is abelian.
> 2. $Ind\_N^G \pi$ is irreducible, if the cardinality of the $G$ orbit $\pi$ has the same cardinality as $G/N$.
> 3. More generally, the cardinality of representations contained in $I\_N^G \pi$ is given by $C(\pi) /N$ for the centralizers $C(\pi) = \{ g \in G : \pi^g =\pi\}$.
> 
> 
> 


1 is obvious by Schur's lemma: Assume $I\_N^G \pi = \bigoplus \sigma^{\oplus m\_\sigma}$ then as algebra isomorphisms
$$ End\_G( I\_N^G \pi) = \bigoplus End\_G( \sigma^{\oplus m\_\sigma}) = \bigoplus M(m(\sigma), F).$$


Proof of 2+3 via Mackey formula, Schur's lemma and Frobenius reciprocity: As vectorspace isomorphisms
$$Endo\_G(I\_N^G \pi) = Hom\_N(\pi , Res\, I\_N^G \pi) = \bigoplus\_{g \in G/N} Hom\_N(\pi, \pi^g).$$


