An Introduction to Automorphic Representations: With a view toward trace formulae

Preface
Acknowledgements
Contents
1 Affine Algebraic Groups
	1.1 Introduction
	1.2 Affine schemes
	1.3 Affine group schemes
	1.4 Extension and restriction of scalars
	1.5 Reductive groups
	1.6 Lie algebras
	1.7 Tori
	1.8 Root data
	1.9 Parabolic subgroups
	1.10 Homogeneous spaces
2 Adeles
	2.1 Adeles
	2.2 Adelic points of affine schemes
	2.3 Relationship with restricted direct products
	2.4 Hyperspecial subgroups and models
	2.5 Approximation in affine algebraic groups
	2.6 The adelic quotient
	2.7 Reduction theory
3 Discrete Automorphic  Representations
	3.1 Representations of locally compact groups
	3.2 Haar measures on locally compact groups
	3.3 Gelfand-Pettis integrals
	3.4 Convolution algebras of test functions
	3.5 Haar measures on local fields
	3.6 Haar measures on the points of affine algebraic groups
	3.7 Discrete automorphic representations
	3.8 Decomposition of representations
	3.9 The Fell topology
	3.10 Type I groups
	3.11 Why affine groups?
4 Archimedean Representation Theory
	4.1 The passage between analysis and algebra
	4.2 Smooth vectors
	4.3 Restriction to compact subgroups
	4.4 (mathfrakg,K)-modules
	4.5 Hecke algebras with K-types
	4.6 Infinitesimal characters
	4.7 Classification of (mathfrakg,K)-modules for GL2mathbbR
	4.8 Matrix coefficients
	4.9 The Langlands classification
5 Representations of Totally  Disconnected Groups
	5.1 Totally disconnected groups
	5.2 Smooth functions on td-groups
	5.3 Smooth and admissible representations
	5.4 Contragredients
	5.5 The unramified Hecke algebra
	5.6 Restricted tensor products of modules
	5.7 Flath\'s theorem
	Exercises
6 Automorphic Forms
	6.1 Smooth functions
	6.2 Classical automorphic forms
	6.3 Adelic automorphic forms over number fields
	6.4 Adelic automorphic forms over function fields
	6.5 The cuspidal subspace
	6.6 Discrete automorphic representations revisited
	6.7 From modular forms to automorphic forms
	6.8 Ramanujan\'s Δ-function
7 Unramified Representations
	7.1 Unramified representations
	7.2 The Satake isomorphism for split groups
	7.3 The Langlands dual group
	7.4 Parabolic subgroups of L-groups
	7.5 The Satake isomorphism for unramified groups
	7.6 The principal series
	7.7 Weak global L-packets
8 Non-Archimedean Representation  Theory
	8.1 Introduction
	8.2 Parabolic induction
	8.3 Jacquet modules
	8.4 The Bernstein-Zelevinsky classification
	8.5 Traces, characters, and coefficients
	8.6 Parabolic descent of representations
	8.7 Parabolic descent of orbital integrals
9 The Cuspidal Spectrum
	9.1 Introduction
	9.2 The cuspidal subspace
	9.3 Deduction of the discreteness of the spectrum
	9.4 The basic estimate
	9.5 The function field case
	9.6 Rapidly decreasing functions
	9.7 Cuspidal automorphic forms
10 Eisenstein Series
	10.1 Induced representations
	10.2 Intertwining operators
	10.3 Eisenstein series
	10.4 Constant terms
	10.5 Decomposition of the spectrum
	10.6 Local preparation for isobaric representations
	10.7 Isobaric representations
	10.8 A theorem of Moeglin and Waldspurger
11 Rankin-Selberg L-functions
	11.1 Paths to the construction of automorphic L-functions
	11.2 Generic characters
	11.3 Generic representations
	11.4 Formulae for Whittaker functions
	11.5 Local Rankin-Selberg L-functions
	11.6 Unramified Rankin-Selberg L-functions
	11.7 Global Rankin-Selberg L-functions
	11.8 The nongeneric case
	11.9 The converse theorem
12 Langlands Functoriality
	12.1 The Weil group
	12.2 The Weil-Deligne group and L-parameters
	12.3 The Archimedean Langlands correspondence
	12.4 The local Langlands correspondence for GLn
	12.5 The local Langlands conjecture
	12.6 Global Langlands functoriality
	12.7 Langlands L-functions
	12.8 Algebraic representations
13 Known Cases of Global Langlands Functoriality
	13.1 Introduction
	13.2 Parabolic induction
	13.3 L-maps into general linear groups
	13.4 Base change
	13.5 The strong Artin conjecture
	13.6 The Langlands-Shahidi method
	13.7 Functoriality for the classical groups
	13.8 Endoscopic classification of representations
	13.9 The function field case
	Exercises
14 Distinction and Period Integrals
	14.1 Introduction
	14.2 Distinction in the local setting
	14.3 Global distinction and period integrals
	14.4 Spherical varieties
	14.5 Symmetric subgroups
	14.6 Relationship with the endoscopic classification
	14.7 Period integrals in the Gan-Gross-Prasad setting
	14.8 Necessary conditions for distinction
15 The Cohomology of Locally  Symmetric Spaces
	15.1 Introduction
	15.2 Locally symmetric spaces
	15.3 Local systems
	15.4 (mathfrakg,Kinfty)-cohomology
	15.5 The cohomology of Shimura manifolds
	15.6 The relation to distinction
	15.7 More on (mathfrakg, Kinfty)-cohomology
	15.8 Shimura varieties
16 Spectral Sides of Trace Formulae
	16.1 The automorphic kernel function
	16.2 Relative traces
	16.3 The full expansion of the automorphic kernel
	16.4 Functions with cuspidal image
17 Orbital Integrals
	17.1 A refined study of orbits
	17.2 Group actions, orbits, and stabilizers
	17.3 Classes and quotients
	17.4 Local geometric classes
	17.5 Local relative orbital integrals
	17.6 Torsors
	17.7 Adelic classes
	17.8 Global relative orbital integrals
18 Simple Trace Formulae
	18.1 A brief history of trace formulae
	18.2 A general simple relative trace formula
	18.3 Products of subgroups
	18.4 The simple trace formula
	18.5 The simple twisted trace formula
	18.6 A variant
	18.7 The Petersson-Bruggeman-Kuznetsov formula
	18.8 Kloosterman integrals
	18.9 Sums of Whittaker coefficients
19 Applications of Trace Formulae
	19.1 Existence and comparison
	19.2 The Weyl law
	19.3 The comparison strategy
	19.4 Jacquet-Langlands transfer and base change
	19.5 Twisted endoscopy
	19.6 The interplay of distinction and twisted endoscopy
A Groups Attached to Involutions of Algebras
	A.1 Algebras with involution
	A.2  Split simple algebras
	A.3  Classification of forms
	A.4  Parabolic subgroups
B The Iwasawa Decomposition
	B.1  Introduction
	B.2  Some group schemes
	B.3  The dynamic method
	B.4  Proof of Theorem B.1.1
	B.5  Addenda in the hyperspecial case
C Poisson Summation
	C.1  The standard additive characters
	C.2  Local Schwartz spaces and Fourier transforms
	C.3  Global Schwartz spaces and Poisson summation
D Alternate Conventions Related to Adelic Quotients
	D.1  The quotients
	D.2  Separating by central character
Hints to selected exercises
References
Index