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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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Explicit Jacquet-Langlands correspondence for real reductive groups

The short answer to all of your questions is: Jacquet-Langlands for $GL(n,F)$ is special because $L$-packets for $GL(n,F)$ are singletons. Consequently the kinds of things you are asking about for oth …
Jeffrey Adams's user avatar