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Let $\Gamma$ be a discrete group that is either Fuchsian ($\Gamma \subseteq \text{PSL}(2,\mathbb R)$) or Kleinian ($\Gamma \subseteq \text{PSL}(2,\mathbb{C})$). I have a unitary representationL $$ \Phi : \Gamma \to \operatorname{U}(\mathcal{H}). $$ For concreteness, I have $\mathcal{H} = L^2(\mathbb{H}^n,\mathbb C)$ for $n=2$ (Fuchsian) and $n=3$ (Kleinian). I am interested in the following questions:

  1. Does $\Phi$ decompose as a direct integral of irreducible representations? If so, are factorial representations finite-dimensional?
  2. Does $\Phi$ decompose as a direct integral of factorial representations? If so, are factorial representations finite-dimensional?

I would be interested to know under which additional assumptions ($\Gamma$ is finitely generated, $\Gamma$ is a surface group etc.) are statements 1 and 2 true. I am also interested in the analagous question for discrete subgroups of the isometry group of $\mathbb H^n$ of $n \geq 4$.

I would appreciate if someone can guide me towards relevanr references etc.

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    $\begingroup$ As in your last question, since we’re dealing with a separable Hilbert space, direct integral decomposition into factorial representations is always possible, and, unless $\Gamma$ is virtually abelian, you always have some representation that cannot be decomposed into a direct integral of irreducible representations (and that include finite-dimensional factorial representations). This really has nothing to do with whether the group is Fuchsian or Kleinian or anything else. $\endgroup$
    – David Gao
    Commented Dec 1 at 4:34
  • $\begingroup$ @DavidGao Is it possible to determine whether a given representation can split into a direct integral of irreducible representations? Any techniques for doing this? The group Gamma is not virtually Abelian for most cases of interest. $\endgroup$
    – user82261
    Commented Dec 1 at 4:50
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    $\begingroup$ A given representation is a direct integral of irreducible representations iff the commutant is abelian. $\endgroup$
    – David Gao
    Commented Dec 1 at 5:07
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    $\begingroup$ (Though, if you just want the representation to be decomposed into type I factors, then that’s equivalent to the vNa generated being type I, or equivalently the commutant being type I. That’s when the direct integral decomposition is of finite-dimensional factors and $B(\ell^2)$. If you want to eliminate $B(\ell^2)$, I guess saying the representation is both of type I and is induced by a trace on the group works. This always happens for virtually abelian groups.) $\endgroup$
    – David Gao
    Commented Dec 1 at 5:22
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    $\begingroup$ Dixmier’s book should work. Takesaki’s Theory of Operator Algebras I also has a section on direct integrals, as well as a chapter on type decomposition for von Neumann algebras. $\endgroup$
    – David Gao
    Commented Dec 1 at 7:22

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