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)?



Restriction of irreducible unitary representation to normal subgroup of finite index
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 $gW$. 


