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Let $G$ be a unitary similitude group over $\mathbb{Q}$ (as in the book of Harris-Taylor), $\pi$ an irreducible automorphic representation of $G(\mathbb{A})$. I'm looking for some results on its multiplicity $m(\pi)$ in the space of automorphic forms. Do we know its exact value under certain conditions? e.g. suppose $\pi$ admit a cuspidal base change (in the sense of Cor.VI.2.8 of Harris-Taylor)?

Many thanks.

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    $\begingroup$ In the case where $\pi$ admits a cuspidal base change, I would think that the multiplicity is 1. I am not completely sure because the these Harris-Taylor unitary groups are not the one I ma familiar with, but for the groups I work with Chenevier in my book "Families of Galois Representations and Selmer Groups". I know that's true. $\endgroup$
    – Joël
    Commented Sep 11, 2014 at 21:03

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