iv CONTENTS
4.3. An elementary lifting result for isogenies 50
4.4. Symmetric square functoriality and Πcusp(PGL3)

52
4.5. Tensor product functoriality and Πcusp(PGL4)
o
53
4.6. Λ∗ functorality and Πcusp(PGL5) o 55
Chapter 5. Description of Πdisc(SO7) and Πalg(PGL6)
s
59
5.1. The semisimple Z-group SO7 59
5.2. Parameterization by the infinitesimal character 59
5.3. Endoscopic partition of Πdisc(SO7) 60
5.4. Conclusions 62
Chapter 6. Description of Πdisc(SO9) and Πalg(PGL8) s 65
6.1. The semisimple Z-group SO9 65
6.2. Endoscopic partition of Πw 65
6.3. Conclusions 68
Chapter 7. Description of Πdisc(SO8) and Πalg(PGL8) o 71
7.1. The semisimple Z-group SO8 71
7.2. Endoscopic partition of Πw 71
7.3. Conclusions 74
Chapter 8. Description of Πdisc(G2) 75
8.1. The semisimple definite G2 over Z 75
8.2. Polynomial invariants for G2(Z) G2(R) 75
8.3. Endoscopic classification of Πdisc(G2) 77
8.4. Conclusions 80
Chapter 9. Application to Siegel modular forms 81
9.1. Vector valued Siegel modular forms of level 1 81
9.2. Two lemmas on holomorphic discrete series 81
9.3. An example: the case of genus 3 84
Appendix A. Adams-Johnson packets 87
A.1. Strong inner forms of compact connected real Lie groups 87
A.2. Adams-Johnson parameters 88
A.3. Adams-Johnson packets 90
A.4. Shelstad’s parameterization map 91
Appendix B. The Langlands group of Z and Sato-Tate groups 95
B.1. The locally compact group LZ 95
B.2. Sato-Tate groups 97
B.3. A list in rank n 8 99
Appendix C. Tables 101
Appendix D. The 121 level 1 automorphic representations of SO25 with
trivial coefficients 115
Bibliography 117
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