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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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Generic irreducibility of parabolic induction
See Theorem 6.6.1 in Casselman's notes titled "Introduction to the theory of admissible representations of $p$-adic groups", available from https://www.math.ubc.ca/~cass/research/pdf/p-adic-book.pdf
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Gelfand pair and double coset decomposition
$K\pi^\lambda K$ has a transitive right action of $K$.
The stabilizer of $K\pi^\lambda$ for this action is $K\cap \pi^{-\lambda}K\pi^\lambda$.
Thus, $K\pi^\lambda K = \coprod_x K\pi^\lambda x$ as $x$ …