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### Contents

1959a Automorphic forms on a semisimple Lie group | 1 |

1960a Supplement to Some results on differential equations | 49 |

1960b Differential equations and semisimple Lie groups | 57 |

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### Common terms and phrases

abelian algebraic group Amer analytic function analytic subgroup assertion follows assume automorphism Cartan involution Cartan subalgebra Cartan subgroup clear commutes compact subset completely invariant complex numbers connected component continuous function converges uniformly corollary of Lemma corresponding defined over Q denote the set differential operator dim g Euclidean measure exists a number finite number Fix a point following result follows from Lemma follows immediately GL(n Haar measure holomorphic function implies induction hypothesis integer invariant distribution irreducible isomorphism Iwasawa decomposition Lemma 17 Lemma 26 Lemma 42 Let G linear function Math Moreover notation open neighborhood open subset orthogonal polynomial function positive number positive roots proof of Lemma proof of Theorem reductive respectively roots of g semisimple Lie algebra semisimple Lie group shows Siegel domain subalgebra of g subgroup of G subspace Supp suppose verify Weyl group zero