Cohomological induction and unitary representations /Anthony W. Knapp and David A. Vogan, Jr.

Includes bibliographies and index.

880-01 5. Abstract Construction6. Hecke Algebras for Pairs (g, K); II. THE CATEGORY C(g, K); 1. Functors P and I; 2. Properties of P and I; 3. Constructions within C(g, K); 4. Special Properties of P and I in Examples; 5. Mackey Isomorphisms; 6. Derived Functors of P and I; 7. Standard Resolutions; 8. Koszul Resolution as a Complex; 9. Reduction of Exactness for the Koszul Resolution; 10. Exactness in the Abelian Case; III. DUALITY THEOREM; 1. Easy Duality; 2. Statement of Hard Duality; 3. Complexes for Computing Pj and I^j; 4. Hard Duality as a K Isomorphism; 5. Proof of g Equivariance in Case (i).

6. Motivation for g Equivariance in Case (ii)7. Proof of g Equivariance in Case (ii); 8. Proof of Hard Duality in the General Case; IV. REDUCTIVE PAIRS; 1. Review of Cartan-Weyl Theory; 2. Cartan-Weyl Theory for Disconnected Groups; 3. Reductive Groups and Reductive Pairs; 4. Cartan Subpairs; 5. Finite-Dimensional Representations; 6. Parabolic Subpairs; 7. Harish-Chandra Isomorphism; 8. Infinitesimal Character; 9. Kostant's Theorem; 10. Casselman-Osborne Theorem; 11. Algebraic Analog of Bott-Borel-Weil Theorem; V. COHOMOLOGICAL INDUCTION; 1. Setting; 2. Effect on Infinitesimal Character.

3. Preliminary Lemmas4. Upper Bound on Multiplicities of K Types; 5. An Euler-Poincaré Principle for K Types; 6. Bottom-Layer Map; 7. Vanishing Theorem; 8. Fundamental Spectral Sequences; 9. Spectral Sequences for Analysis of K Types; 10. Hochschild-Serre Spectral Sequences; 11. Composite P Functors and I Functors; VI. SIGNATURE THEOREM; 1. Setting; 2. Hermitian Dual and Signature; 3. Hermitian Duality Relative to P and I; 4. Statement of Signature Theorem; 5. Comparison of Shapovalov Forms on K and G; 6. Preservation of Positivity from L)"K to K.

7. Signature Theorem for K Badly DisconnectedVII. TRANSLATION FUNCTORS; 1. Motivation and Examples; 2. Generalized Infinitesimal Character; 3. Chevalley's Structure Theorem for Z(g); 4. Z(l) Finiteness of u Homology and Cohomology; 5. Invariants in the Symmetric Algebra; 6 . Kostant's Theory of Harmonics; 7. Dixmier-Duflo Theorem; 8 . Translation Functors; 9. Integral Dominance; 10. Overview of Preservation of Irreducibility; 11. Details of Irreducibility; 12. Nonvanishing of Certain Translation Functors; 13. Application to (g, K) Modules with K Connected.

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