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Let $\pi$ be an automorphic subrepresentation of a reductive group $G$. Here by this, I mean an irreducible representation realized in a subspace of the space of automorphic forms on $G$.

Let $m_\pi$ be the multiplicity of $\pi$ in the space of automorphic forms. Is it in general true that $m_\pi<\infty$? Of course, this is true if $\pi$ is cuspidal or residual. But I would like to know if it is true in general. Also I would like to know where the proof is written even for the cuspidal or residual case.

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    $\begingroup$ The only discrete spectrum is cuspidal or made from residues of cuspidal-data Eisenstein series. In the continuous spectrum, the same continuous spectrum can occur over and over, but not as genuine subrepresentations. What do you mean to ask about "multiplicities" in the latter case? $\endgroup$ Aug 3, 2014 at 14:21
  • $\begingroup$ What I mean is this: Let $(\pi, V_\pi)$ be an irreducible, where $V_\pi$ is the space of $\pi$ such that $V_\pi\subseteq\mathcal{A}(G)$. Then how many other copies of $\pi$ can occur as a subrepresentation of $\mathcal{A}(G)$? Only finitely many or could be infinitely many? $\endgroup$
    – Windi
    Aug 3, 2014 at 17:16

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