Introduction to automorphic representations
Last updated: October 20, 2014
Disclaimer: These notes have been stitched together from lecture
notes that I wrote for graduate classes that I taught at Rutgers-Newark
and at Tel Aviv University, with relatively few subsequent edits. Since
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I Basic notions: Algebraic geometry
1 The language of algebraic geometry: from rings to spaces
1.2 Informal discussion
1.3 Subspaces and (radical) ideals
1.5 Non-reduced rings, non-radical ideals
1.7 The maximal spectrum, and the Zariski topology
2 Affine schemes
2.1 The category of affine schemes
2.2 The underlying space of an affine scheme
2.4 Localization at a prime
3 Sheaves and schemes
4 Noetherian rings
4.2 Recollection of definitions and basic properties
4.3 Primary decomposition and associated ideals
5 Noetherian rings of dimension one
5.1 UFDs and PIDs
5.2 Normal and regular domains
5.3 Local domains
5.4 Language: Dedekind rings, discrete valuation rings; and their properties
6 Various notions of “smoothness”
6.1 Normality and factoriality
6.2 The Zariski cotangent space and regularity
6.3 Differentials and derivations
6.4 Smooth morphisms
6.5 Jacobian criterion and k-smoothness
6.6 Formal smoothness and Hensel’s lemma
6.7 Examples from number theory
7 Divisors and line bundles
7.1 Weil divisors
7.2 Cartier divisors and the Picard group
II Basic notions: Representation Theory
8 Representations of topological groups
8.2 Examples; G-spaces and the regular representation
8.3 Discussion of the continuity condition
9 Representations without topology, discrete groups, finite-dimensional constructions
9.1 Representations on vector spaces without topology
9.2 Discrete groups; the group algebra
9.3 Finite dimensional representations – various constructions
9.3.1 Direct sums
9.3.2 Tensor products
9.3.3 Inner products
10 Representations of finite groups
10.1 Local finiteness
10.3 Schur’s lemma
10.4 The regular representation
10.5 Matrix coefficients
10.6 Exhaustion of CpHq
10.10 Examples: the character tables of S4,S5,A5 etc.
10.11 Irreducible representations of products of groups
11 Representations of compact groups
11.2 Spectral theorems
11.3 Convolution operators
11.4 Peter–Weyl theorems
11.5 Compactness of convolution by continuous measures
11.6 Proof of the main theorems
12 Algebraic groups and Lie groups
12.1 Lie groups, group schemes, algebraic groups
12.2 Extension and restriction of scalars
12.3 From smooth schemes to smooth manifolds
12.4 Open and closed subgroups of Lie groups
12.5 Compact Lie groups are algebraic
13 Lie algebras
13.3 The Lie algebra of a Lie or algebraic group
13.4 Exponential map
13.5 Proof of Cartan’s theorem
13.6 Morphisms of groups and morphisms of Lie algebras
14 Finite-dimensional representations of 𝔰𝔩2pℂq and of general semisimple Lie algebras
14.1 The Lie algebra 𝔰𝔩2pℂq, and a central element
14.2 Highest weight vectors
14.3 Semisimplicity (complete reducibility)
14.4 General 𝔤
15 Structure of general (finite dimensional) Lie algebras
16 Structure of semisimple Lie algebras
16.1 Jordan decomposition in 𝔤𝔩.
16.2 Derivations and the Jordan decomposition
16.3 Cartan subalgebras
16.4 Root decomposition, semisimple case
16.5 Conjugacy of Borel subalgebras and the universal Cartan: statements
16.6 The scheme of Borel subgroups
16.7 Positive roots and standard Borel subgroups
17 Verma modules and the category .
17.1 Verma modules
17.2 The category .
17.3 The case of 𝔰𝔩2, and application.
17.4 Localization with respect to 𝔷p𝔤q
18 The Chevalley and Harish-Chandra isomorphisms
19 Commutative C- and Von Neumann algebras
19.1 Basic definitions
19.2 Invertible elements and characters of a commutative C-algebra
19.3 The Gelfand transform
19.4 Commutative W-algebras
20 General C-algebras and their states
20.1 Corollaries of the Gelfand-Naimark theorem
20.2 Positive elements
20.3 Positive functionals
20.5 Positivity and normal states for W-algebras
20.6 Universal representations
20.7 Projections in a W-algebra
III Algebraic groups and their automorphic quotients
21 Basic notions
21.2 Basic notions
21.3 Homogeneous spaces
21.4 Diagonalizable groups
21.5 Reductive groups
21.6 Root systems and root data
21.6.1 Root systems
21.6.2 Root data
21.6.3 Weyl chambers and based root data
21.7 Parabolic subgroups
22 Structure and forms over a non-algebraically closed field.
22.1 Restriction of scalars
22.4 Forms of reductive groups
22.5 The dual group
22.6 Basic examples
22.6.2 Inner forms of GLn.
22.6.3 Unitary groups.
23 Brauer groups, Galois cohomology.
23.2 Central simple algebras.
23.3 Abelian and non-abelian Galois cohomology
23.4 Basic and important facts of Galois cohomology
23.5 Reciprocity for global Brauer groups.
23.6 The Hasse principle.
24 Recollection of class field theory.
24.2 Local class field theory.
24.3 Global class field theory.
24.4 Hilbert symbols.
24.5 The classical formulation.
24.6 Chebotarev density.
24.7 The dual formulation; Weil groups; Dirichlet characters.
25 The automorphic space.
25.2 The automorphic quotient.
25.3 The additive group
25.4 The multiplicative group
25.5 The general linear group
25.6 Weak and strong approximation.
25.7 Reduction theory for GLn over ℚ
25.8 Arithmetic subgroups, Siegel sets.
25.9 The classical and the adelic picture.
25.10 Genus and class number.
26 Tamagawa numbers.
26.2 Differential forms and measures.
26.3 Global measures
26.4 The Tamagawa measure for reductive groups
26.5 The Tamagawa number of a reductive group
26.6 Picard groups of algebraic groups
26.7 The work of Siegel on quadratic forms
IV Automorphic representations
27 Basic representation theory of real and p-adic groups.
27.2 Continuous representations.
27.3 Continuous representations of compact groups.
27.4 Finite-dimensional representations of Lie groups.
27.4.1 The unitarian trick
27.4.2 Weights; the Cartan decomposition.
27.4.3 Highest weight theory
27.5 Infinite-dimensional representations of Lie groups.
27.5.2 Smooth and K-finite vectors.
27.5.3 Harish-Chandra modules.
27.6 Representations of l-groups: the Hecke algebra.
28 Automorphic forms and the Hecke algebra.
28.1 The case of SL2 – classical approach.
28.2 Representation-theoretic approach.
28.3 Automorphic forms: precise definition (classical).
28.4 Hecke operators
28.5 Adelic formulation. Definition of automorphic representations.
28.6 The unitary spectrum of SL2pℝq; holomorphic modular forms.
29 The Satake isomorphism and automorphic L-functions.
29.1 The tensor product theorem and unramified representations.
29.2 The Satake isomorphism.
29.3 Sketch of proof of the Satake isomorphism.
29.4 Automorphic L-functions.
30 The Langlands conjectures and arithmetic.
30.1 Weil groups and Weil-Deligne groups.
30.2 Local Langlands Conjecture
30.3 The Global Langlands Conjecture
31 Complex representations of p-adic groups.
31.3 Smooth representations
31.4 Induction, restriction.
31.5 Intertwining operators
A Some category theory
A.1 Some category theory
A.2 Some universal objects
A.3 Abelian categories
A.3.1 Axiom 1
A.3.2 Axiom 2
A.3.4 Axiom 3
A.3.6 Axiom 4
A.3.7 Monomorphisms and epimorphisms
A.3.8 Simple objects
A.3.9 The splitting lemma
A.3.10 Jordan–Hölder theorem