Contents 0. Introduction 1 1. Notation 13 1. 0-stable parabolic subalgebra 13 2. good range, fair range 15 3. cohomological parabolic induction 16 4. results from Zuckerman and Vogan - 16 5. results from Harish-Chandra and Oshima-Matsuki 17 2. Main results 19 1. G = Spfaq) 19 2. main theorem for G = Sp(p, q) 20 3. G = U(p, q) 22 4. main theorem for G = U(p, q) 23 5. G = SO0(p, q) 25 6. main theorem for G = SOo(p, q) 26 7. list and figures of various conditions on parameters 28 8. remarks 33 3. Further notations and preliminary results 35 1. Jantzen-Zuckerman's translation functor 35 2. induction by stages 36 3. definition of A (A o A ) 36 4. A{\ A7) and derived functor modules 37 5. some symbols 40 4. Some explicit formulas on K multiplicities 42 1. preliminaries 42 2. some alternating polynomials 48 3. result in quaternionic case 52 4. result in complex case 54 5. result in real case 56 6. some auxiliary lemmas 58 iii

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