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 on $G$.

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)?

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)?