In this memoir, for quasi-split classical groups over p-adic fields, we determine the L-packets consisting of simple supercuspidal representations and their corre-sponding L-parameters, under the assumption that p is not equal to 2. The key is an explicit computation of characters of simple supercuspidal representations and the endoscopic character relation, which is a characterization of the local Langlands correspondence for quasi-split classical groups.

Chapter 1. Introduction 1

Chapter 2. Simple supercuspidal representations of classical groups 11

2.1. Iwahori subgroups 11

2.2. General construction of simple supercuspidal representations 17

2.3. The case of twisted GL_2n 18

2.4. The case of Sp_2n 20

2.5. The case of SO_2n 21

2.6. The case of SO_2n+2 23

2.7. The case of SO_2n+2 26

Chapter 3. Characters of simple supercuspidal representations 29

3.1.The case of twisted GL_2n 30

3.2. The case of SP_2n 33

3.3. The case of SO(_2n)~μ 39

3.4. The case of SO_2n+2

3.5. The case of SO(_2n+2)~ur

Chapter 4. Arthur's local classification theorem 43

Chapter 5. Simple supercuspidal L-packet of Sp_2n 51

5.1. Adjoint orbits of simple supercuspidal representations of Sp_2n 51

5.2. Formal degrees of simple supercuspidal representations 53

5.3. Simple supercuspidal L-packets of Sp_2n 56

Chapter 6. Transfer factors at affine generic elements 59

6.1. Parametrization of conjugacy classes 59

6.2. Norm correspondence 62

6.3. Choice of Whittaker data and the invariant ηG 63

6.4. Normalized transfer factors 66

6.5. Waldspurger's formula for transfer factors 67

6.6. Affine generic data of SO(_2n)~μ 68

6.7. The case of (twisted GL_2n,SO(_2n)~μ) 72

6.8. The case of (Sp_2n, SO(_2n)~μ) 76

6.9. The case of (SO_2n+2,SO(_2n)~μ x SO(_2)~μ) 78

6.10. The case of (SO(_2n+2)~ur,SO(_2n)~μ x SO(_2)~μ) 83

Chapter 7. Endoscopic lifting of simple supercuspidal L-packets of SO(_2n)~μ 87

7.1. Depth bound for the lifted representations of GL_2n+1 88

7.2. Simple supercuspidal L-packet of SO(_2n)~μ 96

7.3. Endoscopic lifting from SO(_2n)~μ to GL_2n 100

Chapter 8. Simple supercuspidal L-packet of SO_2n+2 and SO(_2n+2)~ur 107

8.1. Construction of simple supercuspidal L-packets with ζ=1 107

8.2. Construction of simple supercuspidal L-packets with ζ=-1 117

Chapter 9. Application:Formal degree conjecture for simple supercuspidal L-packets for some bad primes 119

9.1. Formal degree conjecture of Hiraga-Ichino-Ikeda 119

9.2. Outline of the proof of the formal degree conjecture 121

9.3. The case where p does not divide 2n 123

9.4. Some wildly ramified extensions of p-adic fields 128

9.5. Some representation theory of finite Heisenberg groups 136

9.6. The second case where p divides 2n 137

Appendix A. Kloosterman sums and Gauss sums 143

Appendix B. Depth of simple supercuspidal representations 147

B.1. Terminologies from Bruhat-Tits theory 147

B.2. Minimal positivity of depth of simple supercuspidal representations 149

Appendix C. Spinor twists for L-packets of even orthogonal groups 155

Bibliography 159

中科院文献情报中心