The Langlands Classification and Irreducible Characters for Real Reductive Groups

This monograph explores the geometry of the local Langlands conjecture. The conjecture predicts a parametrizations of the irreducible representations of a reductive algebraic group over a local field in terms of the complex dual group and the Weil-Deligne group. For p-adic fields, this conjecture has not been proved; but it has been refined to a detailed collection of (conjectural) relationships between p-adic representation theory and geometry on the space of p-adic representation theory and geometry on the space of p-adic Langlands parameters. This book provides and introduction to some modern geometric methods in representation theory. It is addressed to graduate students and research workers in representation theory and in automorphic forms.

Inhalt

1. Introduction.- 2. Structure theory: real forms.- 3. Structure theory: extended groups and Whittaker models.- 4. Structure theory: L-groups.- 5. Langlands parameters and L-homomorphisms.- 6. Geometric parameters.- 7. Complete geometric parameters and perverse sheaves.- 8. Perverse sheaves on the geometric parameter space.- 9. The Langlands classification for tori.- 10. Covering groups and projective representations.- 11. The Langlands classification without L-groups.- 12. Langlands parameters and Cartan subgroups.- 13. Pairings between Cartan subgroups and the proof of Theorem 10.4.- 14. Proof of Propositions 13.6 and 13.8.- 15. Multiplicity formulas for representations.- 16. The translation principle, the Kazhdan-Lusztig algorithm, and Theorem 1.24.- 17. Proof of Theorems 16.22 and 16.24.- 18. Strongly stable characters and Theorem 1.29.- 19. Characteristic cycles, micro-packets, and Corollary 1.32.- 20. Characteristic cycles and Harish-Chandra modules.- 21. The classification theorem and Harish-Chandra modules for the dual group.- 22. Arthur parameters.- 23. Local geometry of constructible sheaves.- 24. Microlocal geometry of perverse sheaves.- 25. A fixed point formula.- 26. Endoscopic lifting.- 27. Special unipotent representations.- References.