Introduction to automorphic representations (DRAFT!)

Yiannis Sakellaridis

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 the beginning of October 2014 I have started editing them again, slowly, and the date of the latest update is the one you see on the first page. They have not been carefully checked for correctness, and they are very incomplete. Read at your own risk!

Meanwhile, by reading them, you agree to the following rules:

1.
You will not print them out, except for small parts. Paper is too precious to waste on such an incomplete, constantly changing manuscript!
2.
You will let me know of any mathematical mistakes that you encounter. I also welcome any other suggestions/ideas/questions. My email address is: sakellar@rutgers.edu.

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Contents

0 Introduction
I  Basic notions: Algebraic geometry
1 The language of algebraic geometry: from rings to spaces
 1.1 References
 1.2 Informal discussion
 1.3 Subspaces and (radical) ideals
 1.4 Morphisms
 1.5 Non-reduced rings, non-radical ideals
 1.6 Nullstellensatz
 1.7 The maximal spectrum, and the Zariski topology
 1.8 Irreducibility
2 Affine schemes
 2.1 The category of affine schemes
 2.2 The underlying space of an affine scheme
 2.3 Localization
 2.4 Localization at a prime
3 Sheaves and schemes
4 Noetherian rings
 4.1 References
 4.2 Recollection of definitions and basic properties
 4.3 Primary decomposition and associated ideals
 4.4 Dimension
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
 8.1 Definitions
 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
  9.3.4 Duals
10 Representations of finite groups
 10.1 Local finiteness
 10.2 Unitarity
 10.3 Schur’s lemma
 10.4 The regular representation
 10.5 Matrix coefficients
 10.6 Exhaustion of CpHq
 10.7 Corollaries
 10.8 Characters
 10.9 Orthogonality
 10.10 Examples: the character tables of S4,S5,A5 etc.
 10.11 Irreducible representations of products of groups
11 Representations of compact groups
 11.1 Unitarity
 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
 11.7 Characters
 11.8 Examples
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
 13.1 Definitions
 13.2 Poincaré-Birkhoff-Witt
 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 𝔰𝔩2pq and of general semisimple Lie algebras
 14.1 The Lie algebra 𝔰𝔩2pq, 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 O.
 17.1 Verma modules
 17.2 The category O.
 17.3 The case of 𝔰𝔩2, and application.
 17.4 Localization with respect to 𝔷p𝔤q
 17.5 Characters
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
 20.4 States
 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
 21.1 References
 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
  21.6.4 Automorphisms
 21.7 Parabolic subgroups
22 Structure and forms over a non-algebraically closed field.
 22.1 Restriction of scalars
 22.2 Structure
 22.3 Forms
 22.4 Forms of reductive groups
 22.5 The dual group
 22.6 Basic examples
  22.6.1
  22.6.2 Inner forms of GLn.
  22.6.3 Unitary groups.
23 Brauer groups, Galois cohomology.
 23.1 References.
 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.
 24.1 References.
 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.
 25.1 References.
 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
 25.8 Arithmetic subgroups, Siegel sets.
 25.9 The classical and the adelic picture.
 25.10 Genus and class number.
26 Tamagawa numbers.
 26.1 References:
 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.
 27.1 References
 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.
  27.5.1 Admissibility
  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 SL2pq; 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
 30.4 Functoriality
31 Complex representations of p-adic groups.
 31.1 references
 31.2 Introduction
 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.2.1
  A.2.2
  A.2.3
  A.2.4
  A.2.5
  A.2.6
 A.3 Abelian categories
  A.3.1 Axiom 1
  A.3.2 Axiom 2
  A.3.3
  A.3.4 Axiom 3
  A.3.5
  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
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