Contents

Chapter 1. Introduction 1

1.1. A counting problem 1

1.2. Motivations 2

1.3. The main result 3

1.4. Langlands-Sato-Tate groups 4

1.5. The symplectic-orthogonal alternative 5

1.6. Case-by-case description, examples in low motivic weight 6

1.7. Generalizations 11

1.8. Methods and proofs 11

1.9. Application to Borcherds even lattices of rank 25

and determinant 2 13

1.10. A level 1, non-cuspidal, tempered automorphic representation of

GL28 over Q with weights 0, 1, 2, · · · , 27 15

Chapter 2. Polynomial invariants of finite subgroups of compact connected

Lie groups 17

2.1. The setting 17

2.2. The degenerate Weyl character formula 18

2.3. A computer program 19

2.4. Some numerical applications 20

2.5. Reliability 22

2.6. A check: the harmonic polynomial invariants of a Weyl group 23

Chapter 3. Automorphic representations of classical groups : review of

Arthur’s results 25

3.1. Classical semisimple groups over Z 25

3.2. Discrete automorphic representations 27

3.3. The case of Chevalley and definite semisimple Z-groups 28

3.4. Langlands parameterization of Πdisc(G) 30

3.5. Arthur’s symplectic-orthogonal alternative 31

3.6. The symplectic-orthogonal alternative for polarized algebraic regular

cuspidal automorphic representations of GLn over Q 32

3.7. Arthur’s classification: global parameters 36

3.8. The packet Π(ψ) of a ψ ∈ Ψglob(G) 37

3.9. The character εψ of Cψ 41

3.10. Arthur’s multiplicity formula 43

Chapter 4. Determination of Πalg(PGLn) ⊥ for n ≤ 5 47

4.1. Determination of Πcusp(PGL2)

⊥

47

4.2. Determination of Πalg(PGL4) s 47

iii