A local trace formula for the Gan-Gross-Prasad conjecture for unitary groups: the Archimedean case

Introduction
Chapter 1. Preliminaries
1.1. General notation and conventions
1.2. Reminder of norms on algebraic varieties
1.3. A useful lemma
1.4. Common spaces of functions
1.5. Harish-Chandra Schwartz space
1.6. Measures
1.7. Spaces of conjugacy classes and invariant topology
1.8. Orbital integrals and their Fourier transforms
1.9. (G,M)-families
1.10. Weighted orbital integrals
Chapter 2. Representations
2.1. Smooth representations, elliptic regularity
2.2. Unitary and tempered representations
2.3. Parabolic induction
2.4. Normalized intertwining operators
2.5. Weighted characters
2.6. Matricial Paley-Wiener theorem and Plancherel-Harish-Chandra theorem
2.7. Elliptic representations and the space X(G)
Chapter 3. Harish-Chandra descent
3.1. Invariant analysis
3.2. Semi-simple descent
3.3. Descent from the group to its Lie algebra
3.4. Parabolic induction of invariant distributions
Chapter 4. Quasi-characters
4.1. Quasi-characters when F is p-adic
4.2. Quasi-characters on the Lie algebra for F=R
4.3. Local expansions of quasi-characters on the Lie algebra when F=R
4.4. Quasi-characters on the group when F=R
4.5. Functions c
4.6. Homogeneous distributions on spaces of quasi-characters
4.7. Quasi-characters and parabolic induction
4.8. Quasi-characters associated to tempered representations and Whittaker datas
Chapter 5. Strongly cuspidal functions
5.1. Definition, first properties
5.2. Weighted orbital integrals of strongly cuspidal functions
5.3. Spectral characterization of strongly cuspidal functions
5.4. Weighted characters of strongly cuspidal functions
5.5. The local trace formulas for strongly cuspidal functions
5.6. Strongly cuspidal functions and quasi-characters
5.7. Lifts of strongly cuspidal functions
Chapter 6. The Gan-Gross-Prasad triples
6.1. Hermitian spaces and unitary groups
6.2. Definition of GGP triples
6.3. The multiplicity m()
6.4. H"026E30F G is a spherical variety, good parabolic subgroups
6.5. Some estimates
6.6. Relative weak Cartan decompositions
6.7. The function H"026E30F G
6.8. Parabolic degenerations
Chapter 7. Explicit tempered intertwinings
7.1. The -integral
7.2. Definition of L
7.3. Asymptotics of tempered intertwinings
7.4. Explicit intertwinings and parabolic induction
7.5. Proof of Theorem 7.2.1
7.6. A corollary
Chapter 8. The distributions J and J`39`42`"613A``45`47`"603ALie
8.1. The distribution J
8.2. The distribution J`39`42`"613A``45`47`"603ALie
Chapter 9. Spectral expansion
9.1. The theorem
9.2. Study of an auxiliary distribution
9.3. End of the proof of Theorem 9.1.1
Chapter 10. The spectral expansion of J`39`42`"613A``45`47`"603ALie
10.1. The affine subspace
10.2. Conjugation by N
10.3. Characteristic polynomial
10.4. Characterization of '
10.5. Conjugacy classes in '
10.6. Borel subalgebras and '
10.7. The quotient '(F)/H(F)
10.8. Statement of the spectral expansion of J`39`42`"613A``45`47`"603ALie
10.9. Introduction of a truncation
10.10. Change of truncation
10.11. End of the proof of Theorem 10.8.1
Chapter 11. Geometric expansions and a formula[2pt] for the multiplicity
11.1. Some spaces of conjugacy classes
11.2. The linear forms m`39`42`"613A``45`47`"603Ageom and m`39`42`"613A``45`47`"603Ageom`39`42`"613A``45`47`"603ALie
11.3. Geometric multiplicity and parabolic induction
11.4. Statement of three theorems
11.5. Equivalence of Theorem 11.4.1 and Theorem 11.4.2
11.6. Semi-simple descent and the support of J`39`42`"613A``45`47`"603Aqc-m`39`42`"613A``45`47`"603Ageom
11.7. Descent to the Lie algebra and equivalence of Theorem 11.4.1 and Theorem 11.4.3
11.8. A first approximation of J`39`42`"613A``45`47`"603Aqc`39`42`"613A``45`47`"603ALie-m`39`42`"613A``45`47`"603Ageom`39`42`"613A``45`47`"603ALie
11.9. End of the proof
Chapter 12. An application to the[2pt] Gan-Gross-Prasad conjecture
12.1. Strongly stable conjugacy classes, transfer between pure inner forms and the Kottwitz sign
12.2. Pure inner forms of a GGP triple
12.3. The local Langlands correspondence
12.4. The theorem
12.5. Stable conjugacy classes inside (G,H)
12.6. Proof of Theorem 12.4.1
Appendix A. Topological vector spaces
A.1. LF spaces
A.2. Vector-valued integrals
A.3. Smooth maps with values in topological vector spaces
A.4. Holomorphic maps with values in topological vector spaces
A.5. Completed projective tensor product, nuclear spaces
Appendix B. Some estimates
B.1. Three lemmas
B.2. Asymptotics of tempered Whittaker functions for general linear groups
B.3. Unipotent estimates
Bibliography
List of notations