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assume automorphism Cartan subalgebra Cartan subgroup choose cohomology compact complex composition series compute conjugate corresponding data for G define dominant element equivalent finite dimensional representation follows Harish-Chandra highest weight Homº hypothesis imaginary root induction infinitesimal character injective injective resolution integer isomorphism K-finite K-type Langlands Lie algebra maximally split Cartan non-zero nonsingular notation obvious orthogonal parabolic subalgebra parabolic subgroup parity condition positive root system principal series representation Proof Proposition prove Q. E. D. Corollary Q. E. D. Lemma reducibility regular character representation of G root vector satisfies the parity short exact sequence simple root ſº spectral sequence split Cartan subgroup subgroup of G submodule subquotient subspace Suppose G Theorem tion unitary Weyl group write zero