ISBN-10:
0521195608
ISBN-13:
9780521195607
Pub. Date:
08/31/2010
Publisher:
Cambridge University Press
Pseudo-reductive Groups

Pseudo-reductive Groups

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Product Details

ISBN-13: 9780521195607
Publisher: Cambridge University Press
Publication date: 08/31/2010
Series: New Mathematical Monographs Series , #17
Pages: 554
Product dimensions: 6.20(w) x 9.00(h) x 1.40(d)

About the Author

Brian Conrad is a Professor in the Department of Mathematics at Stanford University.

Ofer Gabber is a Directeur de Recherches CNRS at the Institut des Hautes �tudes Scientifiques (IH�S).

Gopal Prasad is Raoul Bott Professor of Mathematics at the University of Michigan.

Table of Contents

Introduction xi

Terminology, conventions, and notation xix

Part I Constructions, examples, and structure theory 1

1 Overview of pseudo-reductivity 3

1.1 Comparison with the reductive case 3

1.2 Elementary properties of pseudo-reductive groups 11

1.3 Preparations for the standard construction 16

1.4 The standard construction and examples 25

1.5 Main result 34

1.6 Weil restriction and fields of definition 36

2 Root groups and root systems 43

2.1 Limits associated to 1 -parameter subgroups 43

2.2 Pseudo-parabolic subgroups 59

2.3 Root groups in pseudo-reductive groups 67

2.4 Representability of automorphism functors 78

3 Basic structure theory 86

3.1 Perfect normal subgroups of pseudo-reductive groups 86

3.2 Root datum for pseudo-reductive groups 94

3.3 Unipotent groups associated to semigroups of roots 99

3.4 Bruhat decomposition and Levi subgroups 114

3.5 Classification of pseudo-parabolic subgroups 130

Part II Standard presentations and their applications 147

4 Variation of (G′, k′/k, T′, C) 149

4.1 Absolutely simple and simply connected fibers 149

4.2 Uniqueness of (G′, k′/k) 154

5 Ubiquity of the standard construction 162

5.1 Main theorem and central extensions 162

5.2 Properties of standardness and standard presentations 170

5.3 A standardness criterion 179

6 Classification results 191

6.1 The A1-case away from characteristic 2 192

6.2 Types A2 and G2 away from characteristic 3 198

6.3 General cases away from characteristics 2 and 3 203

Part III General classification and applications 215

7 The exotic constructions 217

7.1 Calculations in characteristics 2 and 3 217

7.2 Basic exotic pseudo-reductive groups 228

7.3 Algebraic and arithmetic aspects of basic exotic pseudo-reductive groups 240

8 Preparations for classification in characteristics 2 and 3 255

8.1 Further properties of basic exotic pseudo-reductive groups 255

8.2 Exceptional and exotic pseudo-reductive groups 260

9 The absolutely pseudo-simple groups in characteristic 2 279

9.1 TypeA1 280

9.2 Root groups and birational group laws 290

9.3 Construction of absolutely pseudo-simple groups with a non-reduced root system 299

9.4 Classification of absolutely pseudo-simple groups with a non-reduced root system 318

10 General case 342

10.1 Factors with non-reduced root system and the generalized standard construction 342

10.2 Classification via generalized standard groups 351

11 Applications 358

11.1 Maximal tori in pseudo-reductive groups 358

11.2 Pseudo-semisimplicity 364

11.3 Unirationality 368

11.4 Structure of root groups and pseudo-parabolic subgroups 372

Part IV Appendices 389

A Background in linear algebraic groups 391

A.1 Review of definitions 392

A.2 Some results from the general theory 398

A.3 Frobenius morphisms and non-affine groups 402

A.4 Split reductive groups: Existence, Isomorphism, and Isogeny Theorems 407

A.5 Weil restriction generalities 422

A.6 Groups without Levi subgroups 441

A.7 Lie algebras and Weil restriction 446

A.8 Lie algebras and groups of multiplicative type 457

B Tits' work on unipotent groups in nonzero characteristic 473

B.1 Subgroups of vector groups 473

B.2 Wound unipotent groups 479

B.3 The cckp-kernel 482

B.4 Torus actions on unipotent groups 485

C Rational conjugacy in connected groups 494

C.1 Pseudo-completeness 494

C.2 Conjugacy results in the smooth affine case 504

C.3 Split unipotent subgroups of pseudo-reductive groups 511

C.4 Beyond the smooth affine case 519

References 525

Index 527

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