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An automorphic form is a well-behaved function from a topological group $G$ to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup $\Gamma \subset G$ of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.

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How badly can strong multiplicity one fail in the theory of automorphic representations?

They are in the same Arthur packet. The A-packets are designed exactly to answer this question. I think the answers to questions 1,2,3 are "no", "yes conjecturally", and "no": see book of Adams Barb …
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