Restriction of irreducible unitary representation to normal subgroup of finite index Let $G$ be a Lie group (or more generally a locally compact group), let $N$ be a closed and normal subgroup of $G$ of finite index.  Let $H$ be an infinite dimensional complex Hilbert space, and let $\pi$ be an irreducible unitary representation of $G$ on $H$.
Now consider the restriction $\pi|_N$ of $\pi$ to $N$.  Is it true that $\pi|_N$ is the finite sum of irreducible unitary representations?  Perhaps someone could provide an indication of the proof or a reference?
ADDED COMMENT:
Or how about a more basic question:  Must $(\pi|_N, H)$ have a (discretely occuring) irreducible subrepresentation?  Or even just an irreducible quotient (as would exist for a finitely generated algebraic representation)?
 A: I like this question! 
Restricted to the finite index subgroup $N$, the representation $\pi$ splits into a direct sum of irreducible representations. 
I could not see an easy proof of this, but the proof goes along the following lines. 
Suppose $I$ is a totally  ordered indexing set and for each $i\in I$, $W_i$ is an $N$ invariant non-zero (closed) subspace of $H$ such that $i<j$ implies $W_i\supset W_j$. Then I claim that the intersection $\cap _{i\in I}W_i$ is a closed non-zero $N$ invariant subspace. Let $P_i$ be the projection map from $H$ onto $W_i$. This map is $N$-equivariant. Consider the finite sum $p_i= \sum _{g\in G/N} gP_ig^{-1}$. Being a $G$ invariant self adjoint bounded operator, by Schur's lemma, $p_i$ is a scalar $c_iI$ for some $c_i\geq 0$. The scalar $c_i\geq 1$:  if $v\in W_i$ has norm one, then $c_i=(p_iv,v)\geq (P_iv,v)=1$.
On the other hand, the  $W_i$ form  a decreasing family. Suppose $w\in H$. The net of numbers $(P_iw,w)$ decreases to $(Pw,w)$ where $P$ is the projection to the intersection $\cap W_i$. (This last statement needs a proof, which is tedious but routine. One has to replace the family $(P_iw,w)$ by a countable subfamily whose lim inf is the lower limit of the family, and then argue that the lower limit is $(Pw,w)$). 
From the last two paragraphs, for any $w\in H$ of norm one,  we have $c_i=(c_iw,w)=\sum _{g\in G/N}(gP_ig^{-1}w,w)$ decreases to $\sum _{g\in G/N}(gPg^{-1}w,w)$. Since $c_i\geq 1$, it follows that the projection $P$ is non-zero. Hence the intersection $\cap _{i\in I}W_i$ is non-zero. 
By the Hausdorff maximality principle, there exists a minimal $N$ invariant non-zero closed subspace $W$ in $H$. This must necessarily be irreducible. The finite sum $\sum _{g\in G/N}gW$ being non-zero and $G$ invariant, is dense in $H$. Each $gW$ is also $N$-irreducible since $N$ is normal. It follows that $H$ is a direct sum of possibly a smaller collection of $gW$.  
