## Representations of real reductive Lie groups |

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### Contents

Preliminaries | 1 |

4 Finite dimensional representations | 26 |

SL2 IR | 36 |

Copyright | |

31 other sections not shown

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6-stable abelian assume automorphism Cartan subalgebra Cartan subgroup choose coherent family cohomology compact complex composition series compute Conjecture conjugate corresponding data for G decomposition define denote differential dominant element equivalent finite dimensional representation finite length follows formula Harish-Chandra Harish-Chandra module highest weight Horn imaginary root infinitesimal character injective injective resolution integer K-finite K-type Lie algebra lowest K-type maximally split Cartan multiplicity non-zero noncompact imaginary nonsingular notation obvious occurs orthogonal parabolic subalgebra parabolic subgroup parity condition positive root system principal series representation Proof Proposition 5.3.3 prove Q.E.D. Corollary Q.E.D. Definition Q.E.D. Lemma R+(y real root reducible regular character representation of G root a e root vector satisfies the parity short exact sequence simple root space spectral sequence split Cartan subgroup subgroup of G submodule subquotient subspace Suppose G Theorem tion unitary Weyl group write zero