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Let $R$ be real numbers and consider an irreducible unitary representation (\pi,V) of $GL_n(R)$ in some Hilbert space $V$, now $GL_{n-1}(R)$ embeds in $GL_n(R)$ on the left upper diagonal block.

Now I wanna ask properties of the restrictions of $V$ to $GL_{n-1}$. This representation is not irreducible, so we may ask is it possible to say something about the closed invariant subspaces? Does this restriction have a multiplicity free decomposition?

Note that we can ask the same question for other irreducible Banach or Frechet representation of $GL_n$, or replacing real numbers by complex or p-adic numbers.

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In the case of $GL_n$ of a $p$-adic field, this question is related to the theory of derivatives, due to Bernstein and Zelevinsky. You may want to look at their papers on the representation theory of $p$-adic $GL_n$ (which are beautiful in any case). –  Emerton Mar 14 '10 at 19:23

2 Answers 2

Though Peter's answer addresses the finite-dimensional representation theory, I believe that the question asks about the unitary representations on Hilbert spaces, and more general irreps on Banach and Frechet spaces.

This question has been the subject of much recent work by Avraham Aizenbud, Dmitry Gourevitch, Steve Rallis, Gerard Schiffmann, and Eitan Sayag. In particular, Aizenbud and Gourevitch prove the following in their paper "Multiplicity One Theorem for $(GL_{n+1}(R), GL_n(R))$":

Let $F = R$ or $F = C$. Let $\pi$ and $\tau$ be irreducible admissible smooth Fr\'echet representations of $GL_{n+1}(F)$ and $GL_n(F)$, respectively. Then $$dim \left( Hom_{GL_n(F)}(\pi, \tau) \right) \leq 1.$$

This paper is on the ArXiv, and now published in Selecta, according to Aizenbud's webpage.

Zhu and Binyong have also proved this, I believe. The result has also been proven for irreducible smooth repreesentations of $GL_{n+1}(F)$ and $GL_n(F)$, when $F$ is a $p$-adic field by Aizenbud-Gourevitch-Rallis-Schiffmann.

Considering the smooth Fr\'echet case should suffice for the case of unitary representations on Hilbert spaces, I believe, by considering the subspace of smooth vectors and Garding's theorem. I'd guess it would also work for Banach space representations, but I'm not an expert on these analytic things.

It's important to note that semisimplicity may be lost when one restricts smooth representations in these settings -- so their theorem says something about occurrences of quotients after restriction. It's important to be careful about the meaning of "multiplicity-free" in these situations.

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If you use complex numbers instead of real numbers, it is true that the restriction is multiplicity free. This is an important fact, and is used to construct the Gelfand-Zetlin basis for V. This is discussed is Fulton and Harris' book "representation theory", section 25.3. They also discuss generalizations to other classical algebras

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