## Automorphic forms on GL (2) |

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### Contents

Local Theory 1 Well representations | 1 |

Representations of GL2F In the nonarchlmedean case | 23 |

The principal series for nonarcnimedean fields | 92 |

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### Common terms and phrases

absolutely convergent absolutely cuspidal analytically continued archimedean assertion associated automorphic forms bounded in vertical Cartan subgroup choose commutes compact subset constituent contains continuous functions contragredient cosets defined detg direct sum elementary idempotent elements equivalent extension of F finite dimensional finite number finite set following lemma Fourier transform function field functional equation functions f Haar measure holomorphic infinite invariant subspace irreducible admissible representation irreducible representation isomorphism largest ideal Lemma Lie algebra linear transformation locally constant locally constant function matrices Moreover non-archimedean non-negative integer non-trivial non-zero number field obtain orthogonal paragraph polynomial proof Proposition prove quasi-character of F quotient real number replaced representation of G restriction right translations satisfies scalar Schur orthogonality relations space of functions sufficiently large Suppose theorem trivial unitary unramified vector verify vertical strips zero