Ebook: 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.

In the case of real groups, the predicted parametrizations of representations was proved by Langlands himself. Unfortunately, most of the deeper relations suggested by the p-adic theory (between real representation theory and geometry on the space of real Langlands parameters) are not true. The purposed of this book is to redefine the space of real Langlands parameters so as to recover these relationships; informally, to do "Kazhdan-Lusztig theory on the dual group". The new definitions differ from the classical ones in roughly the same way that Deligne’s definition of a Hodge structure differs from the classical one.

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.

---

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.

In the case of real groups, the predicted parametrizations of representations was proved by Langlands himself. Unfortunately, most of the deeper relations suggested by the p-adic theory (between real representation theory and geometry on the space of real Langlands parameters) are not true. The purposed of this book is to redefine the space of real Langlands parameters so as to recover these relationships; informally, to do "Kazhdan-Lusztig theory on the dual group". The new definitions differ from the classical ones in roughly the same way that Deligne’s definition of a Hodge structure differs from the classical one.

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.

---

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.

In the case of real groups, the predicted parametrizations of representations was proved by Langlands himself. Unfortunately, most of the deeper relations suggested by the p-adic theory (between real representation theory and geometry on the space of real Langlands parameters) are not true. The purposed of this book is to redefine the space of real Langlands parameters so as to recover these relationships; informally, to do "Kazhdan-Lusztig theory on the dual group". The new definitions differ from the classical ones in roughly the same way that Deligne’s definition of a Hodge structure differs from the classical one.

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.

Content:
Front Matter....Pages i-xii
Introduction....Pages 1-27
Structure theory: real forms....Pages 28-40
Structure theory: extended groups and Whittaker models....Pages 41-46
Structure theory: L-groups....Pages 47-54
Langlands parameters and L-homomorphisms....Pages 55-63
Geometric parameters....Pages 64-81
Complete geometric parameters and perverse sheaves....Pages 82-97
Perverse sheaves on the geometric parameter space....Pages 98-104
The Langlands classification for tori....Pages 105-112
Covering groups and projective representations....Pages 113-119
The Langlands classification without L-groups....Pages 120-138
Langlands parameters and Cartan subgroups....Pages 139-146
Pairings between Cartan subgroups and the proof of Theorem 10.4....Pages 147-156
Proof of Propositions 13.6 and 13.8....Pages 157-166
Multiplicity formulas for representations....Pages 167-174
The translation principle and the Kazhdan-Lusztig algorithm....Pages 175-188
Proof of Theorems 16.22 and 16.24....Pages 189-204
Strongly stable characters and Theorem 1.29....Pages 205-211
Characteristic cycles, micro-packets, and Corollary 1.32....Pages 212-221
Characteristic cycles and Harish-Chandra modules....Pages 222-233
The classification theorem and Harish-Chandra modules for the dual group....Pages 234-238
Arthur parameters....Pages 239-247
Local geometry of constructible sheaves....Pages 248-251
Microlocal geometry of perverse sheaves....Pages 252-265
A fixed point formula....Pages 266-274
Endoscopic lifting....Pages 275-294
Special unipotent representations....Pages 295-310
Back Matter....Pages 311-320

---

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.

In the case of real groups, the predicted parametrizations of representations was proved by Langlands himself. Unfortunately, most of the deeper relations suggested by the p-adic theory (between real representation theory and geometry on the space of real Langlands parameters) are not true. The purposed of this book is to redefine the space of real Langlands parameters so as to recover these relationships; informally, to do "Kazhdan-Lusztig theory on the dual group". The new definitions differ from the classical ones in roughly the same way that Deligne’s definition of a Hodge structure differs from the classical one.

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.

Content:
Front Matter....Pages i-xii
Introduction....Pages 1-27
Structure theory: real forms....Pages 28-40
Structure theory: extended groups and Whittaker models....Pages 41-46
Structure theory: L-groups....Pages 47-54
Langlands parameters and L-homomorphisms....Pages 55-63
Geometric parameters....Pages 64-81
Complete geometric parameters and perverse sheaves....Pages 82-97
Perverse sheaves on the geometric parameter space....Pages 98-104
The Langlands classification for tori....Pages 105-112
Covering groups and projective representations....Pages 113-119
The Langlands classification without L-groups....Pages 120-138
Langlands parameters and Cartan subgroups....Pages 139-146
Pairings between Cartan subgroups and the proof of Theorem 10.4....Pages 147-156
Proof of Propositions 13.6 and 13.8....Pages 157-166
Multiplicity formulas for representations....Pages 167-174
The translation principle and the Kazhdan-Lusztig algorithm....Pages 175-188
Proof of Theorems 16.22 and 16.24....Pages 189-204
Strongly stable characters and Theorem 1.29....Pages 205-211
Characteristic cycles, micro-packets, and Corollary 1.32....Pages 212-221
Characteristic cycles and Harish-Chandra modules....Pages 222-233
The classification theorem and Harish-Chandra modules for the dual group....Pages 234-238
Arthur parameters....Pages 239-247
Local geometry of constructible sheaves....Pages 248-251
Microlocal geometry of perverse sheaves....Pages 252-265
A fixed point formula....Pages 266-274
Endoscopic lifting....Pages 275-294
Special unipotent representations....Pages 295-310
Back Matter....Pages 311-320
....