Let G be the F-points of a classical group defined over a p-adic field F of characteristic 0. We classify the irreducible unitarizable representation ofG that are subquotients ofthe parabolic induction ofcuspidal representations ofLevi subgroup of corank at most 3 in G.

Chapter 1. Introduction 1

Chapter 2. Notation and Preliminary Results 9

2.1. General linear groups 9

2.2. Classifications of admissible duals of general linear groups 10

2.3. Classical groups – basic definitions 11

2.4. Twisted Hopf algebra structure 12

2.5. Some formulas for M~∗ 14

2.6. Langlands classification for classical groups ([49], [11], [26], [43], [72]) 14

2.7. Irreducible subquotients of induced representations of classical groups 15

2.8. Involution 15

2.9. Reducibility point and generalized Steinberg representations 16

2.10. Representations of segment type 16

2.11. Jordan blocks 18

2.12. Induction of GL-type 20

2.13. Technical lemma on irreducibility 20

2.14. Distinguished irreducible subquotient in induced representation 21

2.15. Some well-known ways of obtaining unitarizability 24

2.16. Reduction of unitarizability to the weakly real case 25

2.17. Computing irreducible subquotients 26

Chapter 3. Unitarizability in the Critical Case (Corank 1 and 2) 27

3.1. Extreme cases 27

3.2. Tempered representations in critical case, corank ≤ 3 28

3.3. Composition series in critical case, corank one 30

3.4. Composition series in critical case, corank two 31

Chapter 4. Unitarizability in the Critical Case (Corank 3, α > 1) 37

4.1. x = (α, α + 1, α + 2) and α ≥1/2 37

4.2. x = (α, α + 1, α + 1) and α ≥ 1/2 37

4.3. x = (α, α, α + 1) and α ≥ 1 38

4.4. x = (α, α, α) and α ≥ 1 40

4.5. x = (α − 1, α, α + 1) and α > 1 40

4.6. x = (α − 1, α, α) and α > 1 44

4.7. x = (α − 1, α − 1, α) and α > 1 46

4.8. x = (α − 2, α − 1, α) and α ≥ 2 49

Chapter 5. Remaining Cases for α= 1/2 and α = 1 53

5.1. x = (0, 1, 2) and α = 1 53

5.2. x = (0, 1, 1) and α = 1 57

5.3. x = (0, 0, 1) and α = 1 59

5.4. x = (1/2,1/2,3/2 ) and α =1/2 60

5.5. x = (1/2,1/2,1/2 ) and α =1/2 65

Chapter 6. The Case α = 0 67

6.1. x = (0, 1, 2) and α = 0 67

6.2. x = (0, 1, 1) and α = 0 74

6.3. x = (0, 0, 1) and α = 0 75

6.4. x = (0, 0, 0) and α = 0 77

Chapter 7. Introductory Remarks on Unitarizability and Corank 2 79

7.1. Corank 1 79

7.2. Corank 2 80

7.3. General principles related to graphic interpretations (cf. §2.15) 80

7.4. Proof of Proposition 7.2 for α ≥ 1 80

7.5. Proof of Proposition 7.2 for α =1/2 81

7.6. Proof of Proposition 7.2 for α = 0 82

Chapter 8. Unitarizability in Corank 3 85

8.1. One-parameter complementary series 85

8.2. Regular components, unitarizability 86

8.3. Two-parameter complementary series – slanted hyperplanes case 88

8.4. Two-parameter complementary series – level hyperplanes case 94

8.5. Three-parameter complementary series 98

8.6. Conclusion 101

8.7. Conjectures 104

Chapter 9. Unitarizability in Mixed Case for Corank ≤ 3 105

9.1. Jantzen decomposition 105

9.2. Preservation of unitarizability by decomposition in corank ≤ 3 106

Appendix A. The Arthur Packet of L(ν~α ρ, ν~(α−1) ρ; δ(ν~α ρ; σ))

by Colette Mœglin 111

A.1. The representations 111

A.2. The parameters 111

A.3. The result 111

Appendix B. Jacquet Module of L(ν~α ρ, ν~(α−1) ρ; δ(ν~α ρ; σ)) 113

Bibliography 117

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