Representations of Real Reductive Lie Groups

assume automorphism B₁ Cartan subalgebra Cartan subgroup choose cohomology compact complex composition series compute conjugate corresponding data for G define Definition dominant element finite dimensional representation finite length follows functor Harish-Chandra highest weight imaginary root infinitesimal character injective integer isomorphism K-finite K-type Langlands Lemma Lie algebra maximally split Cartan non-zero nonsingular notation obvious occurs orthogonal parabolic subalgebra parabolic subgroup parity condition positive root system principal series representation Proof prove Q.E.D. Lemma real root reducible regular character representation of G root vector roots of h satisfies the parity short exact sequence simple root spectral sequence split Cartan subgroup subgroup of G submodule subspace Suppose G Theorem tion trivial unitary Weyl group write zero α α α ε ξ+λ