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 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
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