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.

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 S_{4},S_{5},A_{5} 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 PoincarÃ©-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 𝔰𝔩_{2}pℂq and of general semisimple Lie algebras

14.1 The Lie algebra 𝔰𝔩_{2}pℂq, 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 𝔰𝔩_{2}, 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 GL_{n}.

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 GL_{n} 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.

26.1 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 SL_{2} – 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 SL_{2}pℝq; 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–HÃ¶lder theorem

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 S

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 PoincarÃ©-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 𝔰𝔩

14.1 The Lie algebra 𝔰𝔩

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 𝔰𝔩

17.4 Localization with respect to 𝔷p𝔤q

17.5 Characters

18 The Chevalley and Harish-Chandra isomorphisms

19 Commutative C

19.1 Basic definitions

19.2 Invertible elements and characters of a commutative C

19.3 The Gelfand transform

19.4 Commutative W

20 General C

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

20.6 Universal representations

20.7 Projections in a W

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 GL

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 GL

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 SL

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 SL

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–HÃ¶lder theorem

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