Representation Theory of Algebraic Groups

Representation theory of algebraic groupsFrom Wikipedia, the free encyclopedia

Contents

1 (B, N) pair 11.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Properties of groups with a BN pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Adelic algebraic group 32.1 Ideles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Tamagawa numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 History of the terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Algebraic group 53.1 Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Algebraic subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Coxeter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Algebraic torus 74.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2 Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.4 Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.5 Arithmetic invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Approximation in algebraic groups 105.1 Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

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5.3 Formal denitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6 Arason invariant 126.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

7 Arithmetic group 137.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

8 Borel subgroup 148.1 Parabolic subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148.2 Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

9 Borelde Siebenthal theory 169.1 Connected subgroups of maximal rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169.2 Maximal connected subgroups of maximal rank . . . . . . . . . . . . . . . . . . . . . . . . . . . 169.3 Closed subsystems of roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189.4 Applications to symmetric spaces of compact type . . . . . . . . . . . . . . . . . . . . . . . . . . 189.5 Applications to hermitian symmetric spaces of compact type . . . . . . . . . . . . . . . . . . . . . 209.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

10 BottSamelson variety 2310.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

11 Bruhat decomposition 2411.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2411.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2411.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.4 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

12 Cartan subgroup 2612.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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12.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2612.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2612.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

13 Chevalleys structure theorem 2713.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

14 Cohomological invariant 2814.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2814.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2814.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

15 Complexication (Lie group) 3015.1 Universal complexication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

15.1.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3015.1.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3015.1.3 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3015.1.4 Injectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

15.2 Chevalley complexication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3115.2.1 Hopf algebra of matrix coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3115.2.2 Invariant theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

15.3 Decompositions in the Chevalley complexication . . . . . . . . . . . . . . . . . . . . . . . . . . 3215.3.1 Cartan decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3215.3.2 Gauss decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3215.3.3 Bruhat decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3315.3.4 Iwasawa decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

15.4 Complex structures on homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3415.5 Noncompact real forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

15.5.1 Involutions of simply connected compact Lie groups . . . . . . . . . . . . . . . . . . . . . 3515.5.2 Conjugations on the complexication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3615.5.3 Cartan decomposition in a real form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3615.5.4 Iwasawa decomposition in a real form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

15.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3715.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3715.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

16 Cuspidal representation 4116.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4116.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

17 Diagonalizable group 4317.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

18 Dieudonn module 44

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18.1 Dieudonn rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4418.2 Dieudonn modules and groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4518.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4518.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

19 Dierential algebraic group 4619.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

20 Dierential Galois theory 4720.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4720.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4720.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4720.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

21 E6 (mathematics) 4921.1 Real and complex forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4921.2 E6 as an algebraic group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5021.3 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

21.3.1 Dynkin diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5021.3.2 Roots of E6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5021.3.3 Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5321.3.4 Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

21.4 Important subalgebras and representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5421.5 E6 polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5421.6 Chevalley and Steinberg groups of type E6 and 2E6 . . . . . . . . . . . . . . . . . . . . . . . . . . 5421.7 Importance in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5521.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5621.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

22 E7 (mathematics) 5822.1 Real and complex forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5822.2 E7 as an algebraic group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5922.3 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

22.3.1 Dynkin diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5922.3.2 Root system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5922.3.3 Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6122.3.4 Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

22.4 Important subalgebras and representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6222.4.1 E7 Polynomial Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

22.5 Chevalley groups of type E7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6322.6 Importance in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6322.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6322.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

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22.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

23 E8 (mathematics) 6523.1 Basic description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6523.2 Real and complex forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6523.3 E8 as an algebraic group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6623.4 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6623.5 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6723.6 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6723.7 E8 root system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

23.7.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6723.7.2 Dynkin diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6823.7.3 Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6923.7.4 Simple roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7023.7.5 Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.7.6 E8 root lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.7.7 Simple subalgebras of E8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

23.8 Chevalley groups of type E8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.9 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7123.10Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7323.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7323.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7423.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

24 F4 (mathematics) 7624.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

24.1.1 Dynkin diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7624.1.2 Weyl/Coxeter group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7624.1.3 Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7624.1.4 F4 lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7624.1.5 Roots of F4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7724.1.6 F4 polynomial invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

24.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7924.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7924.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

25 Fixed-point subgroup 81

26 Formal group 8226.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8226.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8326.3 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8326.4 The logarithm of a commutative formal group law . . . . . . . . . . . . . . . . . . . . . . . . . . 84

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26.5 The formal group ring of a formal group law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8426.6 Formal group laws as functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8526.7 The height of a formal group law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8526.8 Lazard ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8526.9 Formal groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8626.10LubinTate formal group laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8726.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

27 Fundamental lemma (Langlands program) 8927.1 Motivation and history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8927.2 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8927.3 Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9027.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9027.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9027.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

28 G2 (mathematics) 9228.1 Real forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9228.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

28.2.1 Dynkin diagram and Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9228.2.2 Roots of G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.2.3 Weyl/Coxeter group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.2.4 Special holonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

28.3 Polynomial Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.4 Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.5 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.6 Finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

29 Geometric invariant theory 9629.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9629.2 Mumfords book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9729.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9829.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9929.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

30 Glossary of algebraic groups 10030.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

31 Good ltration 10131.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

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32 Grosshans subgroup 10232.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10232.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

33 Group of Lie type 10333.1 Classical groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10333.2 Chevalley groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10333.3 Steinberg groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10333.4 SuzukiRee groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10433.5 Relations with nite simple groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10533.6 Small groups of Lie type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10533.7 Notation issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10633.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10733.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10733.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

34 Group scheme 10934.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10934.2 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10934.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11034.4 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11134.5 Finite at group schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11134.6 Cartier duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11234.7 Dieudonn modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11234.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

35 Haboushs theorem 11435.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11435.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11435.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

36 HochschildMostow group 11636.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

37 Hyperspecial subgroup 11737.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

38 Inner form 11838.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

39 Iwahori subgroup 11939.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

40 JordanChevalley decomposition 120

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40.1 Decomposition of endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12040.2 Decomposition in a real semisimple Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12040.3 Decomposition in a real semisimple Lie group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12140.4 Counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12140.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

41 KazhdanLusztig polynomial 12241.1 Motivation and history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12241.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12241.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12341.4 KazhdanLusztig conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

41.4.1 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12441.5 Relation to intersection cohomology of Schubert varieties . . . . . . . . . . . . . . . . . . . . . . 12541.6 Generalization to real groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12541.7 Generalization to other objects in representation theory . . . . . . . . . . . . . . . . . . . . . . . 12641.8 Combinatorial theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12641.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12641.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

42 Kempf vanishing theorem 12842.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

43 KneserTits conjecture 12943.1 Fields for which the Whitehead group vanishes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12943.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12943.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

44 Kostant polynomial 13044.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13044.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13044.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13144.4 Steinberg basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13344.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

45 Langs theorem 13545.1 The LangSteinberg theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13545.2 Proof of Langs theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13545.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13645.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

46 Langlands decomposition 13746.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13746.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13746.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

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47 Lattice (discrete subgroup) 13847.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13947.2 Arithmetic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13947.3 S-arithmetic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13947.4 Adelic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13947.5 Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13947.6 Tree lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14047.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14047.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

48 Lazards universal ring 14148.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

49 LieKolchin theorem 14249.1 Triangularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14249.2 Lies theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14249.3 Counter-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14349.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

50 Mirabolic group 14450.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

51 MumfordTate group 14551.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14551.2 MumfordTate conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14651.3 Period conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14651.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14651.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14651.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

52 Observable subgroup 14752.1 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

53 Pseudo-reductive group 14853.1 Examples of pseudo reductive groups that are not reductive . . . . . . . . . . . . . . . . . . . . . 14853.2 Classication and exotic phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14853.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

54 Quasi-split group 15054.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15054.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

55 Radical of an algebraic group 15155.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

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55.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

56 Rational representation 15256.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

57 Reductive group 15357.1 Lie group case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15357.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15357.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15457.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

58 Restricted Lie algebra 15558.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15558.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15558.3 Restricted universal enveloping algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15558.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15558.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

59 Root datum 15759.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15759.2 The root datum of an algebraic group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15759.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

60 Rost invariant 15960.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15960.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

61 Semisimple algebraic group 16161.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16161.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16161.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16161.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

62 Serre group 16262.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16262.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

63 SeveriBrauer variety 16363.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16463.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16463.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

64 Siegel parabolic subgroup 16564.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

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65 Spaltenstein variety 16665.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

66 Special group (algebraic group theory) 167

67 Springer resolution 16867.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

68 Steinberg representation 16968.1 The Steinberg representation of a nite group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16968.2 The Steinberg representation of a p-adic group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16968.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

69 Superstrong approximation 17169.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17169.2 Proofs of superstrong approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17169.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17169.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

70 Taniyama group 17370.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

71 Tannakian category 17471.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17471.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17471.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

72 Thin group (algebraic group theory) 17672.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

73 Unipotent 17773.1 Unipotent algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17773.2 Unipotent radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17773.3 Jordan decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17873.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17873.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

74 Weil conjecture on Tamagawa numbers 17974.1 Tamagawa measure and Tamagawa numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17974.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17974.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18074.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18074.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

75 Weyl module 181

xii CONTENTS

75.1 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

76 Witt vector 18276.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

76.1.1 Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18276.2 Construction of Witt rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18376.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18476.4 Universal Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18476.5 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

76.5.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18576.5.2 Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18576.5.3 Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

76.6 Ring schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18676.7 Commutative unipotent algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18676.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18676.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

77 Wonderful compactication 18777.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

78 tale group scheme 18878.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18878.2 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 189

78.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18978.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19278.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Chapter 1

(B, N) pair

In mathematics, a (B, N) pair is a structure on groups of Lie type that allows one to give uniform proofs of manyresults, instead of giving a large number of case-by-case proofs. Roughly speaking, it shows that all such groups aresimilar to the general linear group over a eld. They were invented by the mathematician Jacques Tits, and are alsosometimes known as Tits systems.

1.1 DenitionA (B, N) pair is a pair of subgroups B and N of a group G such that the following axioms hold:

G is generated by B and N. The intersection, H, of B and N is a normal subgroup of N. The groupW = N/H is generated by a set S of elements wi of order 2, for i in some non-empty set I. If wi is an element of S and w is any element ofW, then wiBw is contained in the union of BwiwB and BwB. No generator wi normalizes B.

The idea of this denition is that B is an analogue of the upper triangular matrices of the general linear groupGLn(K),H is an analogue of the diagonal matrices, and N is an analogue of the normalizer of H.The subgroup B is sometimes called the Borel subgroup, H is sometimes called the Cartan subgroup, and W iscalled theWeyl group. The pair (W,S) is a Coxeter system.The number of generators is called the rank.

1.2 Examples Suppose that G is any doubly transitive permutation group on a set X with more than 2 elements. We let B bethe subgroup of G xing a point x, and we let N be the subgroup xing or exchanging 2 points x and y. Thesubgroup H is then the set of elements xing both x and y, and W has order 2 and its nontrivial element isrepresented by anything exchanging x and y.

Conversely, if G has a (B, N) pair of rank 1, then the action of G on the cosets of B is doubly transitive. SoBN pairs of rank 1 are more or less the same as doubly transitive actions on sets with more than 2 elements.

Suppose thatG is the general linear groupGLn(K) over a eld K. We take B to be the upper triangular matrices,H to be the diagonal matrices, and N to be the monomial matrices, i.e. matrices with exactly one non-zeroelement in each row and column. There are n 1 generators wi, represented by the matrices obtained byswapping two adjacent rows of a diagonal matrix.

1

2 CHAPTER 1. (B, N) PAIR

More generally, any group of Lie type has the structure of a BN-pair.

A reductive algebraic group over a local eld has a BN-pair where B is an Iwahori subgroup.

1.3 Properties of groups with a BN pairThe map taking w to BwB is an isomorphism from the set of elements of W to the set of double cosets of B; this isthe Bruhat decomposition G = BWB.If T is a subset of S then let W(T) be the subgroup of W generated by T : we dene and G(T) = BW(T)B to be thestandard parabolic subgroup for T. The subgroups of G containing conjugates of B are the parabolic subgroups; con-jugates of B are called Borel subgroups (or minimal parabolic subgroups). These are precisely the standard parabolicsubgroups.

1.4 ApplicationsBN-pairs can be used to prove that many groups of Lie type are simple modulo their centers. More precisely, if Ghas a BN-pair such that B is a solvable group, the intersection of all conjugates of B is trivial, and the set of generatorsofW cannot be decomposed into two non-empty commuting sets, then G is simple whenever it is a perfect group. Inpractice all of these conditions except for G being perfect are easy to check. Checking that G is perfect needs someslightly messy calculations (and in fact there are a few small groups of Lie type which are not perfect). But showingthat a group is perfect is usually far easier than showing it is simple.

1.5 References Bourbaki, Nicolas (2002). Lie Groups and Lie Algebras: Chapters 46. Elements of Mathematics. Springer.ISBN 3-540-42650-7. Zbl 0983.17001. The standard reference for BN pairs.

Serre, Jean-Pierre (2003). Trees. Springer. ISBN 3-540-44237-5. Zbl 1013.20001.

Chapter 2

Adelic algebraic group

In abstract algebra, an adelic algebraic group is a semitopological group dened by an algebraic group G over anumber eld K, and the adele ring A = A(K) of K. It consists of the points of G having values in A; the denition ofthe appropriate topology is straightforward only in caseG is a linear algebraic group. In the case ofG an abelian varietyit presents a technical obstacle, though it is known that the concept is potentially useful in connection with Tamagawanumbers. Adelic algebraic groups are widely used in number theory, particularly for the theory of automorphicrepresentations, and the arithmetic of quadratic forms.In case G is a linear algebraic group, it is an ane algebraic variety in ane N-space. The topology on the adelicalgebraic groupG(A) is taken to be the subspace topology in AN , the Cartesian product of N copies of the adele ring.

2.1 IdelesAn important example, the idele group I(K), is the case of G = GL1 . Here the set of ideles (also idles /dlz/)consists of the invertible adeles; but the topology on the idele group is not their topology as a subset of the adeles.Instead, considering that GL1 lies in two-dimensional ane space as the 'hyperbola' dened parametrically by

{(t, t1)},

the topology correctly assigned to the idele group is that induced by inclusion in A2; composing with a projection, itfollows that the ideles carry a ner topology than the subspace topology from A.Inside AN , the product KN lies as a discrete subgroup. This means that G(K) is a discrete subgroup of G(A), also. Inthe case of the idele group, the quotient group

I(K)/K

is the idele class group. It is closely related to (though larger than) the ideal class group. The idele class group is notitself compact; the ideles must rst be replaced by the ideles of norm 1, and then the image of those in the idele classgroup is a compact group; the proof of this is essentially equivalent to the niteness of the class number.The study of the Galois cohomology of idele class groups is a central matter in class eld theory. Characters of theidele class group, now usually called Hecke characters, give rise to the most basic class of L-functions.

2.2 Tamagawa numbersSee also: Weil conjecture on Tamagawa numbers

For more general G, the Tamagawa number is dened (or indirectly computed) as the measure of

G(A)/G(K).

3

4 CHAPTER 2. ADELIC ALGEBRAIC GROUP

Tsuneo Tamagawa's observation was that, starting from an invariant dierential form on G, dened over K, themeasure involved was well-dened: while could be replaced by c with c a non-zero element of K, the productformula for valuations in K is reected by the independence from c of the measure of the quotient, for the productmeasure constructed from on each eective factor. The computation of Tamagawa numbers for semisimple groupscontains important parts of classical quadratic form theory.

2.3 History of the terminologyHistorically the idles were introduced by Chevalley (1936) under the name "lment idal, which is ideal elementin French, which Chevalley (1940) then abbreviated to idle following a suggestion of Hasse. (In these papers healso gave the ideles a non-Hausdor topology.) This was to formulate class eld theory for innite extensions in termsof topological groups. Weil (1938) dened (but did not name) the ring of adeles in the function eld case and pointedout that Chevalleys group of Idealelemente was the group of invertible elements of this ring. Tate (1950) dened thering of adeles as a restricted direct product, though he called its elements valuation vectors rather than adeles.Chevalley (1951) dened the ring of adeles in the function eld case, under the name repartitions. The term adle(short for additive idles, and also a French womans name) was in use shortly afterwards (Jaard 1953) and mayhave been introduced by Andr Weil. The general construction of adelic algebraic groups by Ono (1957) followedthe algebraic group theory founded by Armand Borel and Harish-Chandra.

2.4 References Chevalley, Claude (1936), Gnralisation de la thorie du corps de classes pour les extensions innies.,Journal de Mathmatiques Pures et Appliques (in French) 15: 359371, JFM 62.1153.02

Chevalley, Claude (1940), La thorie du corps de classes, Annals of Mathematics. Second Series 41: 394418, ISSN 0003-486X, JSTOR 1969013, MR 0002357

Chevalley, Claude (1951), Introduction to the Theory of Algebraic Functions of One Variable, MathematicalSurveys, No. VI, Providence, R.I.: American Mathematical Society, MR 0042164

Jaard, Paul (1953), Anneaux d'adles (d'aprs Iwasawa), Sminaire Bourbaki, Secrtariat mathmatique,Paris, MR 0157859

Ono, Takashi (1957), Sur une proprit arithmtique des groupes algbriques commutatifs, Bulletin de laSocit Mathmatique de France 85: 307323, ISSN 0037-9484, MR 0094362

Tate, John T. (1950), Fourier analysis in number elds, and Heckes zeta-functions, Algebraic Number The-ory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305347, ISBN 978-0-9502734-2-6, MR 0217026

Weil, Andr (1938), Zur algebraischen Theorie der algebraischen Funktionen., Journal fr Reine und Ange-wandte Mathematik (in German) 179: 129133, doi:10.1515/crll.1938.179.129, ISSN 0075-4102

2.5 External links Rapinchuk, A.S. (2001), Tamagawa number, inHazewinkel, Michiel, Encyclopedia ofMathematics, Springer,ISBN 978-1-55608-010-4

Chapter 3

Algebraic group

In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that themultiplication and inversion operations are given by regular functions on the variety.In terms of category theory, an algebraic group is a group object in the category of algebraic varieties.

3.1 ClassesSeveral important classes of groups are algebraic groups, including:

Finite groups GL(n, C), the general linear group of invertible matrices over C Jet group Elliptic curves.

Two important classes of algebraic groups arise, that for the most part are studied separately: abelian varieties (the'projective' theory) and linear algebraic groups (the 'ane' theory). There are certainly examples that are neither onenor the other these occur for example in the modern theory of integrals of the second and third kinds such as theWeierstrass zeta function, or the theory of generalized Jacobians. But according to Chevalleys structure theorem anyalgebraic group is an extension of an abelian variety by a linear algebraic group. This is a result of Claude Chevalley:if K is a perfect eld, and G an algebraic group over K, there exists a unique normal closed subgroup H in G, suchthat H is a linear group and G/H an abelian variety.According to another basic theorem, any group in the category of ane varieties has a faithful nite-dimensionallinear representation: we can consider it to be a matrix group over K, dened by polynomials over K and with matrixmultiplication as the group operation. For that reason a concept of ane algebraic group is redundant over a eld we may as well use a very concrete denition. Note that this means that algebraic group is narrower than Lie group,when working over the eld of real numbers: there are examples such as the universal cover of the 22 special lineargroup that are Lie groups, but have no faithful linear representation. A more obvious dierence between the twoconcepts arises because the identity component of an ane algebraic group G is necessarily of nite index in G.When one wants to work over a base ring R (commutative), there is the group scheme concept: that is, a group objectin the category of schemes over R. Ane group scheme is the concept dual to a type of Hopf algebra. There is quitea rened theory of group schemes, that enters for example in the contemporary theory of abelian varieties.

3.2 Algebraic subgroupAn algebraic subgroup of an algebraic group is a Zariski closed subgroup. Generally these are taken to be connected(or irreducible as a variety) as well.

5

6 CHAPTER 3. ALGEBRAIC GROUP

Another way of expressing the condition is as a subgroup which is also a subvariety.This may also be generalized by allowing schemes in place of varieties. The main eect of this in practice, apartfrom allowing subgroups in which the connected component is of nite index > 1, is to admit non-reduced schemes,in characteristic p.

3.3 Coxeter groupsMain article: Coxeter groupFurther information: Field with one element

There are a number of analogous results between algebraic groups and Coxeter groups for instance, the number ofelements of the symmetric group is n! , and the number of elements of the general linear group over a nite eldis the q-factorial [n]q! ; thus the symmetric group behaves as though it were a linear group over the eld with oneelement. This is formalized by the eld with one element, which considers Coxeter groups to be simple algebraicgroups over the eld with one element.

3.4 See also Algebraic topology (object) Borel subgroup Tame group Morley rank CherlinZilber conjecture Adelic algebraic group Glossary of algebraic groups

3.5 Notes

3.6 References Chevalley, Claude, ed. (1958), Sminaire C. Chevalley, 1956-1958. Classication des groupes de Lie al-gbriques, 2 vols, Paris: Secrtariat Mathmatique, MR 0106966, Reprinted as volume 3 of Chevalleys col-lected works.

Humphreys, James E. (1972), Linear Algebraic Groups, Graduate Texts in Mathematics 21, Berlin, New York:Springer-Verlag, ISBN 978-0-387-90108-4, MR 0396773

Lang, Serge (1983), Abelian varieties, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90875-5 Milne, J. S., Ane Group Schemes; Lie Algebras; Lie Groups; Reductive Groups; Arithmetic Subgroups Mumford, David (1970),Abelian varieties, OxfordUniversity Press, ISBN978-0-19-560528-0, OCLC138290 Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics 9 (2nd ed.), Boston, MA:Birkhuser Boston, ISBN 978-0-8176-4021-7, MR 1642713

Waterhouse, William C. (1979), Introduction to ane group schemes, Graduate Texts in Mathematics 66,Berlin, New York: Springer-Verlag, ISBN 978-0-387-90421-4

Weil, Andr (1971), Courbes algbriques et varits abliennes, Paris: Hermann, OCLC 322901

Chapter 4

Algebraic torus

In mathematics, an algebraic torus is a type of commutative ane algebraic group. These groups were named byanalogy with the theory of tori in Lie group theory (see maximal torus). The theory of tori is in some sense oppositeto that of unipotent groups, because tori have rich arithmetic structure but no deformations.

4.1 DenitionGiven a base scheme S, an algebraic torus over S is dened to be a group scheme over S that is fpqc locally isomorphicto a nite product of copies of the multiplicative group schemeGm/S over S. In other words, there exists a faithfullyat map X S such that any point in X has a quasi-compact open neighborhood U whose image is an open anesubscheme of S, such that base change to U yields a nite product of copies of GL,U = Gm/U. One particularlyimportant case is when S is the spectrum of a eld K, making a torus over S an algebraic group whose extension tosome nite separable extension L is a nite product of copies of Gm/L. In general, the multiplicity of this product(i.e., the dimension of the scheme) is called the rank of the torus, and it is a locally constant function on S.If a torus is isomorphic to a product of multiplicative groupsGm/S, the torus is said to be split. All tori over separablyclosed elds are split, and any non-separably closed eld admits a non-split torus given by restriction of scalars over aseparable extension. Restriction of scalars over an inseparable eld extension will yield a commutative group schemethat is not a torus.

4.2 WeightsOver a separably closed eld, a torus T admits two primary invariants. The weight lattice X(T ) is the group ofalgebraic homomorphisms T G, and the coweight lattice X(T ) is the group of algebraic homomorphisms G T. These are both free abelian groups whose rank is that of the torus, and they have a canonical nondegeneratepairingX(T )X(T )! Z given by (f; g) 7! deg(f g) , where degree is the number n such that the compositionis equal to the nth power map on the multiplicative group. The functor given by taking weights is an antiequivalenceof categories between tori and free abelian groups, and the coweight functor is an equivalence. In particular, mapsof tori are characterized by linear transformations on weights or coweights, and the automorphism group of a torusis a general linear group over Z. The quasi-inverse of the weights functor is given by a dualization functor from freeabelian groups to tori, dened by its functor of points as:

D(M)S(X) := Hom(M;Gm;S(X)):

This equivalence can be generalized to pass between groups of multiplicative type (a distinguished class of formalgroups) and arbitrary abelian groups, and such a generalization can be convenient if one wants to work in a well-behaved category, since the category of tori doesn't have kernels or ltered colimits.When a eld K is not separably closed, the weight and coweight lattices of a torus over K are dened as the respectivelattices over the separable closure. This induces canonical continuous actions of the absolute Galois group of K on

7

8 CHAPTER 4. ALGEBRAIC TORUS

the lattices. The weights and coweights that are xed by this action are precisely the maps that are dened over K. Thefunctor of taking weights is an antiequivalence between the category of tori over K with algebraic homomorphismsand the category of nitely generated torsion free abelian groups with an action of the absolute Galois group of K.Given a nite separable eld extension L/K and a torus T over L, we have a Galois module isomorphism

X(ResL/KT ) = IndGKGLX(T ):If T is the multiplicative group, then this gives the restriction of scalars a permutation module structure. Tori whoseweight lattices are permutation modules for the Galois group are called quasi-split, and all quasi-split tori are niteproducts of restrictions of scalars.For a general base scheme S, weights and coweights are dened as fpqc sheaves of free abelian groups on S. Theseprovide representations of fundamental groupoids of the base with respect the fpqc topology. If the torus is locallytrivializable with respect to a weaker topology such as the etale topology, then the sheaves of groups descend to thesame topologies and these representations factor through the respective quotient groupoids. In particular, an etalesheaf gives rise to a quasi-isotrivial torus, and if S is locally noetherian and normal (more generally, geometricallyunibranched), the torus is isotrivial. As a partial converse, a theorem of Grothendieck asserts that any torus of nitetype is quasi-isotrivial, i.e., split by an etale surjection.Given a rank n torus T over S, a twisted form is a torus over S for which there exists a fpqc covering of S for whichtheir base extensions are isomorphic, i.e., it is a torus of the same rank. Isomorphism classes of twisted forms ofa split torus are parametrized by nonabelian at cohomology H1(S;GLn(Z)) , where the coecient group forms aconstant sheaf. In particular, twisted forms of a split torus T over a eld K are parametrized by elements of the Galoiscohomology pointed setH1(GK ; GLn(Z))with trivial Galois action on the coecients. In the one-dimensional case,the coecients form a group of order two, and isomorphism classes of twisted forms of G are in natural bijectionwith separable quadratic extensions of K.Since taking a weight lattice is an equivalence of categories, short exact sequences of tori correspond to short exactsequences of the corresponding weight lattices. In particular, extensions of tori are classied by Ext1 sheaves. Theseare naturally isomorphic to the at cohomology groupsH1(S;HomZ(X(T1); X(T2))) . Over a eld, the extensionsare parametrized by elements of the corresponding Galois cohomology group.

4.3 ExampleLet S be the restriction of scalars of G over the eld extension C/R. This is a real torus whose real points formthe Lie group of nonzero complex numbers. Restriction of scalars gives a canonical embedding of S into GL2, andcomposition with determinant gives an algebraic homomorphism of tori from S to G, called the norm. The kernelof this map is a nonsplit rank one torus called the norm torus of the extension C/R, and its real points form the Liegroup U(1), which is topologically a circle. It has no multiplicative subgroups (equivalently, the weight lattice has nononzero Galois xed points), and such tori are called anisotropic. Its weight lattice is a copy of the integers, with thenontrivial Galois action that sends complex conjugation to the minus one map.

4.4 IsogeniesAn isogeny is a surjective morphism of tori whose kernel is a nite at group scheme. Equivalently, it is an injectionof the corresponding weight lattices with nite cokernel. The degree of the isogeny is dened to be the order of thekernel, i.e., the rank of its structure sheaf as a locally free OS -module, and it is a locally constant function on thebase. One can also dene the degree to be order of the cokernel of the corresponding linear transformation on weightlattices. Two tori are called isogenous if there exists an isogeny between them. An isogeny is an isomorphism if andonly if its degree is one. Note that if S doesn't have a map to Spec Q, then the kernel may not be smooth over S.Given an isogeny f of degree n, one can prove using linear algebra on weights and faithfully at descent that thereexists a dual isogeny g such that gf is the nth power map on the source torus. Therefore, isogeny is an equivalencerelation on the category of tori. T. Ono pointed out that two tori over a eld are isogenous if and only if their weightlattices are rationally equivalent as Galois modules, where rational equivalence means we tensor over Z with Q andget equivalent vector spaces with Galois action. This extends naturally from Galois modules to fpqc sheaves, whereZ and Q are constant sheaves rather than plain groups.

4.5. ARITHMETIC INVARIANTS 9

4.5 Arithmetic invariantsIn his work on Tamagawa numbers, T. Ono introduced a type of functorial invariants of tori over nite separableextensions of a chosen eld k. Such an invariant is a collection of positive real-valued functions fK on isomorphismclasses of tori over K, as K runs over nite separable extensions of k, satisfying three properties:

1. Multiplicativity: Given two tori T1 and T2 over K, fK(T1 T2) = fK(T1) fK(T2)

2. Restriction: For a nite separable extension L/K, fL evaluated on an L torus is equal to fK evaluated on itsrestriction of scalars to K.

3. Projective triviality: If T is a torus over K whose weight lattice is a projective Galois module, then fK(T) = 1.

T. Ono showed that the Tamagawa number of a torus over a number eld is such an invariant. Furthermore, he showedthat it is a quotient of two cohomological invariants, namely the order of the groupH1(Gk; X(T )) = Ext1(T;Gm)(sometimes mistakenly called the Picard group of T, although it doesn't classify G torsors over T), and the order ofthe TateShafarevich group.The notion of invariant given above generalizes naturally to tori over arbitrary base schemes, with functions takingvalues in more general rings. While the order of the extension group is a general invariant, the other two invariantsabove do not seem to have interesting analogues outside the realm of fraction elds of one-dimensional domains andtheir completions.

4.6 See also Torus based cryptography Toric geometry

4.7 References A. Grothendieck, SGA 3 Exp. VIIIX T. Ono, On Tamagawa Numbers T. Ono, On the Tamagawa number of algebraic tori Annals of Mathematics 78 (1) 1963.

Chapter 5

Approximation in algebraic groups

In algebraic group theory, approximation theorems are an extension of the Chinese remainder theorem to algebraicgroups G over global elds k.

5.1 UseThey give conditions for the group G(k) to be dense in a restricted direct product of groups of the form G(ks) for ksa completion of k at the place s. In weak approximation theorems the product is over a nite set of places s, while instrong approximation theorems the product is over all but a nite set of places.

5.2 HistoryEichler (1938) proved strong approximation for some classical groups. Strong approximation was established in the1960s and 1970s, for semisimple simply-connected algebraic groups over global elds. The results for number eldsare due to Kneser (1966) and Platonov (1969); the function eld case, over nite elds, is due to Margulis (1977) andPrasad (1977). In the number eld case Platonov also proved a related a result over local elds called the KneserTitsconjecture.

5.3 Formal denitions and propertiesLet G be a linear algebraic group over a global eld k, and A the adele ring of k. If S is a non-empty nite set ofplaces of k, then we write AS for the ring of S-adeles and AS for the product of the completions ks, for s in the niteset S; thus A = AS AS. For any choice of S, G(k) embeds in G(AS) and G(AS).The question asked in weak approximation is whether the embedding of G(k) in G(AS) has dense image. If the groupG is connected and k-rational, then it satises weak approximation with respect to any set S (Platonov, Rapinchuk1994, p.402). More generally, for any connected group G, there is a nite set T of nite places of k such that Gsatises weak approximation with respect to any set S that is disjoint with T (Platonov, Rapinchuk 1994, p.415). Inparticular, if k is an algebraic number eld then any group G satises weak approximation with respect to the set S =S of innite places.The question asked in strong approximation is whether the embedding of G(k) in G(AS) has dense image, or equiva-lently whether the set

G(k)G(AS)

is a dense subset in G(A). The main theorem of strong approximation (Kneser 1966, p.188) states that a non-solvablelinear algebraic group G over a global eld k has strong approximation for the nite set S if and only if its radical Nis unipotent, G/N is simply connected, and each almost simple component H of G/N has a non-compact componentHs for some s in S (depending on H).

10

5.4. SEE ALSO 11

The proofs of strong approximation depended on the Hasse principle for algebraic groups, which for groups of typeE8 was only proved several years later.Weak approximation holds for a broader class of groups, including adjoint groups and inner forms of Chevalleygroups, showing that the strong approximation property is restrictive.

5.4 See also Superstrong approximation

5.5 References Eichler, Martin (1938), Allgemeine Kongruenzklasseneinteilungen der Ideale einfacher Algebren ber alge-braischen Zahlkrpern und ihre L-Reihen., Journal fr Reine und Angewandte Mathematik (in German) 179:227251, doi:10.1515/crll.1938.179.227, ISSN 0075-4102

Kneser, Martin (1966), Strong approximation, Algebraic Groups and Discontinuous Subgroups (Proc. Sym-pos. Pure Math., Boulder, Colo., 1965), Providence, R.I.: American Mathematical Society, pp. 187196, MR0213361

Margulis, G. A. (1977), Cobounded subgroups in algebraic groups over local elds, Akademija Nauk SSSR.Funkcional'nyi Analiz i ego Priloenija 11 (2): 4557, 95, ISSN 0374-1990, MR 0442107

Platonov, V. P. (1969), The problem of strong approximation and the KneserTits hypothesis for algebraicgroups, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 33: 12111219, ISSN 0373-2436, MR0258839

Platonov, Vladimir; Rapinchuk, Andrei (1994), Algebraic groups and number theory. (Translated from the1991 Russian original by Rachel Rowen.), Pure and Applied Mathematics 139, Boston, MA: Academic Press,Inc., ISBN 0-12-558180-7, MR 1278263

Prasad, Gopal (1977), Strong approximation for semi-simple groups over function elds, Annals of Mathe-matics. Second Series 105 (3): 553572, ISSN 0003-486X, JSTOR 1970924, MR 0444571

Chapter 6

Arason invariant

In mathematics, the Arason invariant is a cohomological invariant associated to a quadratic form of even rank andtrivial discriminant and Cliord invariant over a eld k of characteristic not 2, taking values in H3(k,Z/2Z). It wasintroduced by (Arason 1975, Theorem 5.7).The Rost invariant is a generalization of the Arason invariant to other algebraic groups.

6.1 DenitionSuppose thatW(k) is theWitt ring of quadratic forms over a eld k and I is the ideal of forms of even dimension. TheArason invariant is a group homomorphism from I3 to the Galois cohomology group H3(k,Z/2Z). It is determinedby the property that on the 8-dimensional diagonal form with entries 1, a, b, ab, -c, ac, bc, -abc (the 3-fold Psterforma,b,c) it is given by the cup product of the classes of a, b, c in H1(k,Z/2Z) = k*/k*2. The Arason invariantvanishes on I4, and it follows from the Milnor conjecture proved by Voevodsky that it is an isomorphism from I3/I4to H3(k,Z/2Z).

6.2 References Arason, Jn Kr. (1975), Cohomologische Invarianten quadratischer Formen, J. Algebra (in German) 36 (3):448491, doi:10.1016/0021-8693(75)90145-3, ISSN 0021-8693, MR 0389761, Zbl 0314.12104

Esnault, Hlne; Kahn, Bruno; Levine, Marc; Viehweg, Eckart (1998), The Arason invariant and mod 2algebraic cycles, J. Amer. Math. Soc. 11 (1): 73118, doi:10.1090/S0894-0347-98-00248-3, ISSN 0894-0347, MR 1460391, Zbl 1025.11009

Garibaldi, Skip; Merkurjev, Alexander; Serre, Jean-Pierre (2003), Cohomological invariants in Galois coho-mology, University Lecture Series 28, Providence, RI: American Mathematical Society, ISBN 0-8218-3287-5,MR 1999383, Zbl 1159.12311

Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998), The book of involutions,Colloquium Publications 44, With a preface by J. Tits, Providence, RI: American Mathematical Society, p.436, ISBN 0-8218-0904-0, Zbl 0955.16001

12

Chapter 7

Arithmetic group

In mathematics, an arithmetic group is a subgroup of a linear algebraic group with simple algebraic properties.

7.1 Formal denitionAn arithmetic group (arithmetic subgroup) in a linear algebraic groupG dened over a number eld K is a subgroup of G(K) that is commensurable with G(O), where O is the ring of integers of K. Here two subgroups A and B of agroup are commensurable when their intersection has nite index in each of them. It can be shown that this conditiondepends only on G, not on a given matrix representation of G.

7.2 ExamplesExamples of arithmetic groups include the groups GLn(Z). The idea of arithmetic group is closely related to that oflattice in a Lie group. Lattices in that sense tend to be arithmetic, except in well-dened circumstances.

7.3 HistoryThe exact relationship of arithmetic groups and lattices in Lie groups was established by the work of Margulis onsuperrigidity. The general theory of arithmetic groups was developed by Armand Borel and Harish-Chandra; thedescription of their fundamental domains was in classical terms the reduction theory of algebraic forms.

7.4 References Hazewinkel, Michiel, ed. (2001), Arithmetic group, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

13

Chapter 8

Borel subgroup

In the theory of algebraic groups, aBorel subgroup of an algebraic groupG is amaximal Zariski closed and connectedsolvable algebraic subgroup. For example, in the group GLn (n x n invertible matrices), the subgroup of invertibleupper triangular matrices is a Borel subgroup.For groups realized over algebraically closed elds, there is a single conjugacy class of Borel subgroups.Borel subgroups are one of the two key ingredients in understanding the structure of simple (more generally, reductive)algebraic groups, in Jacques Tits' theory of groups with a (B,N) pair. Here the group B is a Borel subgroup and N isthe normalizer of a maximal torus contained in B.The notion was introduced by Armand Borel, who played a leading role in the development of the theory of algebraicgroups.

8.1 Parabolic subgroups

Subgroups between a Borel subgroup B and the ambient group G are called parabolic subgroups. Parabolic sub-groups P are also characterized, among algebraic subgroups, by the condition thatG/P is a complete variety. Workingover algebraically closed elds, the Borel subgroups turn out to be the minimal parabolic subgroups in this sense.Thus B is a Borel subgroup when the homogeneous space G/B is a complete variety which is as large as possible.For a simple algebraic group G, the set of conjugacy classes of parabolic subgroups is in bijection with the set of allsubsets of nodes of the corresponding Dynkin diagram; the Borel subgroup corresponds to the empty set and G itselfcorresponding to the set of all nodes. (In general each node of the Dynkin diagram determines a simple negativeroot and thus a one-dimensional 'root group' of G---a subset of the nodes thus yields a parabolic subgroup, generatedby B and the corresponding negative root groups. Moreover any parabolic subgroup is conjugate to such a parabolicsubgroup.)

8.2 Lie algebra

For the special case of a Lie algebra g with a Cartan subalgebra h , given an ordering of h , the Borel subalgebra is thedirect sum of h and the weight spaces of g with positive weight. A Lie subalgebra of g containing a Borel subalgebrais called a parabolic Lie algebra.

8.3 See also

Hyperbolic group

14

8.4. REFERENCES 15

8.4 References Gary Seitz (1991). Algebraic Groups. In B. Hartley et al. Finite and Locally Finite Groups. pp. 4570. J. Humphreys (1972). Linear Algebraic Groups. New York: Springer. ISBN 0-387-90108-6. A. Borel (2001). Essays in the History of Lie Groups and Algebraic Groups. Providence RI: AMS. ISBN0-8218-0288-7.

8.5 External links Popov, V.L. (2001), Parabolic subgroup, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

Platonov, V.P. (2001), Borel subgroup, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

Chapter 9

Borelde Siebenthal theory

In mathematics, Borelde Siebenthal theory describes the closed connected subgroups of a compact Lie group thathavemaximal rank, i.e. contain a maximal torus. It is named after the Swiss mathematicians Armand Borel and Jeande Siebenthal who developed the theory in 1949. Each such subgroup is the identity component of the centralizer ofits center. They can be described recursively in terms of the associated root system of the group. The subgroups forwhich the corresponding homogeneous space has an invariant complex structure correspond to parabolic subgroupsin the complexication of the compact Lie group, a reductive algebraic group.

9.1 Connected subgroups of maximal rank

Let G be connected compact Lie group with maximal torus T. Hopf showed that the centralizer of a torus S T isa connected closed subgroup containing T, so of maximal rank. Indeed, if x is in CG(S), there is a maximal toruscontaining both S and x and it is contained in CG(S).[1]

Borel and de Siebenthal proved that the connected closed subgroups of maximal rank are precisely the identity com-ponents of the centralizers of their centers.[2]

Their result relies on a fact from representation theory. The weights of an irreducible representation of a connectedcompact semisimple group K with highest weight can be easily described (without their multiplicities): they areprecisely the saturation under the Weyl group of the dominant weights obtained by subtracting o a sum of simpleroots from . In particular, if the irreducible representation is trivial on the center of K (a nite Abelian group), 0 isa weight.[3]

To prove the characterization of Borel and de Siebenthal, let H be a closed connected subgroup of G containing Twith center Z. The identity component L of CG(Z) contains H. If it were strictly larger, the restriction of the adjointrepresentation of L to H would be trivial on Z. Any irreducible summand, orthogonal to the Lie algebra of H, wouldprovide non-zero weight zero vectors for T / Z H / Z, contradicting the maximality of the torus T / Z in L / Z.[4]

9.2 Maximal connected subgroups of maximal rank

Borel and de Siebenthal classied the maximal closed connected subgroups of maximal rank of a connected compactLie group.The general classication of connected closed subgroups of maximal rank can be reduced to this case, because anyconnected subgroup of maximal rank is contained in a nite chain of such subgroups, each maximal in the next one.Maximal subgroups are the identity components of any element of their center not belonging to the center of thewhole group.The problem of determining the maximal connected subgroups of maximal rank can be further reduced to the casewhere the compact Lie group is simple. In fact the Lie algebra g of a connected compact Lie group G splits as adirect sum of the ideals

16

9.2. MAXIMAL CONNECTED SUBGROUPS OF MAXIMAL RANK 17

g = z g1 gm;where z is the center and the other factors gi are simple. If T is a maximal torus, its Lie algebra t has a correspondingsplitting

t = z t1 tm;where ti is maximal abelian in gi . If H is a closed connected of G containing T with Lie algebra h , the complexi-cation of h is the direct sum of the complexication of t and a number of one-dimensional weight spaces, each ofwhich lies in the complexication of a factor gi . Thus if

hi = h \ gi;then

h = z h1 hm:If H is maximal, all but one of the hi 's coincide with gi and the remaining one is maximal and of maximal rank.For that factor, the closed connected subgroup of the corresponding simply connected simple compact Lie group ismaximal and of maximal rank.[5]

Let G be a connected simply connected compact simple Lie group with maximal torus T. Let g be the Lie algebra ofG and t that of T. Let be the corresponding root system. Choose a set of positive roots and corresponding simpleroots 1, ..., n. Let 0 the highest root in gC and write

0 = m11 + +mnnwith mi 1. (The number of mi equal to 1 is equal to |Z| 1, where Z is the center of G.)TheWeyl alcove is dened by

A = fT 2 t : 1(T ) 0; : : : ; n(T ) 0; 0(T ) 1g:lie Cartan shouwed that it is a fundamental domain for the ane Weyl group. If G1 = G / Z and T1 = T / Z, itfollows that the exponential mapping from g to G1 carries 2A onto T1.The Weyl alcove A is a simplex with vertices at

v0 = 0; vi = m1i Xi;

where i(Xj) = ij.The main result of Borel and de Siebenthal is as follows.THEOREM. The maximal connected subgroups of maximal rank in G1 up to conjugacy have the form CG1 (Xi) for mi = 1 CG1(vi) for mi a prime.The structure of the corresponding subgroup H1 can be described in both cases. It is semisimple in the second casewith a system of simple roots obtained by replacing i by 0. In the rst case it is the direct product of the circlegroup generated by Xi and a semisimple compact group with a system of simple roots obtained by omitting i.This result can be rephrased in terms of the extended Dynkin diagram of g which adds an extra node for the highestroot as well as the labelsmi. The maximal subalgebras h of maximal rank are either non-semisimple or semimsimple.The non-semisimple ones are obtained by deleting two nodes from the extended diagram with coecient one. Thecorresponding unlabelled diagram gives the Dynkin diagram semisimple part of h , the other part being a one-dimensional factor. The Dynkin diagrams for the semisimple ones are obtained by removing one node with coecienta prime. This leads to the following possibilities:

18 CHAPTER 9. BORELDE SIEBENTHAL THEORY

An: Ap A n p T (non-semisimple)

Bn: Dn or Bp Dn p (semisimple), Bn T (non-semisimple)

Cn: Cp Cn p (SS), An - T (NSS)

Dn: Dp Dn - p (SS), Dn - T, A- T (NSS)

E6: A1 A5, A2 A2 A2 (SS), D5 T (NSS)

E7: A1 D6, A2 A5, A7 (SS), E6 T (NSS)

E8: D8, A8, A4 A4, E6 A2, E7 A1 (SS)

F4: B4, A2 A2, A1 C3 (SS)

G2: A2, A1 A1 (SS)

All the corresponding homogeneous spaces are symmetric, since the subalgebra is the xed point algebra of an innerautomorphism of period 2, apart fromG2/A2, F4/A2A2, E6/A2A2A2, E7/A2A5 and all the E8 spaces other thanE8/D8 and E8/E7A1. In all these exceptional cases the subalgebra is the xed point algebra of an inner automorphismof period 3, except for E8/A4A4 where the automorphism has period 5. The homogeneous spaces are then calledweakly symmetric spaces.To prove the theorem, note that H1 is the identity component of the centralizer of an element exp T with T in 2 A.Stabilizers increase in moving from a subsimplex to an edge or vertex, so T either lies on an edge or is a vertex. Ifit lies on an edge than that edge connects 0 to a vertex vi with mi = 1, which is the rst case. If T is a vertex vi andmi has a non-trivial factor m, then mT has a larger stabilizer than T, contradicting maximality. So mi must be prime.Maximality can be checked directly using the fact that an intermediate subgroup K would have the same form, so thatits center would be either (a) T or (b) an element of prime order. If the center of H1 is 'T, each simple root with miprime is already a root of K, so (b) is not possible; and if (a) holds, i is the only root that could be omitted with mj= 1, so K = H1. If the center of H1 is of prime order, j is a root of K for mj = 1, so that (a) is not possible; if (b)holds, then the only possible omitted simple root is i, so that K = H1.[6]

9.3 Closed subsystems of rootsA subset 1 is called a closed subsystem if whenever and lie in 1 with + in , then + lies in 1.Two subsystems 1 and 2 are said to be equivalent if ( 1) = 2 for some in W = NG(T) / T, the Weyl group.Thus for a closed subsystem

tC M21

g

is a subalgebra of gC containing tC ; and conversely any such subalgebra gives rise to a closed subsystem. Borel andde Siebenthal classied the maximal closed subsystems up to equivalence.[7]

THEOREM. Up to equivalence the closed root subsystems are given by mi = 1 with simple roots all j with j i or by mi > 1 prime with simple roots 0 and all j with j i.This result is a consequence of the Borelde Siebenthal theorem for maximal connected subgroups of maximal rank.It can also be proved directly within the theory of root systems and reection groups.[8]

9.4 Applications to symmetric spaces of compact typeLet G be a connected compact semisimple Lie group, an automorphism of G of period 2 and G the xed pointsubgroup of . Let K be a closed subgroup of G lying between G and its identity component. The compact homo-geneous space G / K is called a symmetric space of compact type. The Lie algebra g admits a decomposition

9.4. APPLICATIONS TO SYMMETRIC SPACES OF COMPACT TYPE 19

g = k p;where k , the Lie algebra of K, is the +1 eigenspace of and p the 1 eigenspace. If k contains no simple summandof g , the pair ( g , ) is called an orthogonal symmetric Lie algebra of compact type.[9]

Any inner product on g , invariant under the adjoint representation and , induces a Riemannian structure on G /K, with G acting by isometries. Under such an inner product, k and p are orthogonal. G / K is then a Riemanniansymmetric space of compact type.[10]

The symmetric space or the pair ( g , ) is said to be irreducible if the adjoint action of k (or equivalently the identitycomponent of G or K) is irreducible on p . This is equivalent to the maximality of k as a subalgebra.[11]

In fact there is a one-one correspondence between intermediate subalgebras h and K-invariant subspaces p1 of p givenby

h = k p1; p1 = h \ p:Any orthogonal symmetric algebra ( g , ) can be decomposed as an (orthogonal) direct sum of irreducible orthogonalsymmetric algebras.[12]

In fact g can be written as a direct sum of simple algebras

g = Ni=1gi;which are permuted by the automorphism . If leaves an algebra g1 invariant, its eigenspace decomposition coin-cides with its intersections with k and p . So the restriction of to g1 is irreducible. If interchanges two simplesummands, the corresponding pair is isomorphic to a diagonal inclusion of K in K K, with K simple, so is alsoirreducible. The involution just swaps the two factors (x,y)=(y,x).This decomposition of an orthogonal symmetric algebra yields a direct product decomposition of the correspondingcompact symmetric spaceG / K whenG is simply connected. In this case the xed point subgroupG is automaticallyconnected (this is no longer true, even for inner involutions, if G is not simply connected).[13] For simply connectedG, the symmetric space G / K is the direct product of the two kinds of symmetric spaces Gi / Ki or H H / H.Non-simply connected symmetric space of compact type arise as quotients of the sinply connected space G / K bynite Abelian groups. In fact if

G/K = G1/K1 Gs/Ks;let

i = Z(Gi)/Z(Gi) \Kiand let i be the subgroup of i xed by all automorphisms ofGi preserving Ki (i.e. automorphisms of the orthogonalsymmetric Lie algebra). Then

= 1 sis a nite Abelian group acting freely on G / K. The non-simply connected symmetric spaces arise as quotients bysubgroups of . The subgroup can be identied with the fundamental group, which is thus a nite Abelian group.[14]

The classication of compact symmetric spaces or pairs ( g , ) thus reduces to the case whereG is a connected simplecompact Lie group. There are two possibilities: either the automorphism is inner, in which case K has maximalrank and the theory of Borel and de Siebenthal applies; or the automorphism is outer, so that, because preserves amaximal torus, the rank of K is less than the rank ofG and corresponds to an automorphism of the Dynkin diagrammodulo inner automorphisms. Wolf (2010) determines directly all possible in the latter case: they correspond tothe symmetric spaces SU(n)/SO(n), SO(a+b)/SO(a)SO(b) (a and b odd), E6/F4 and E6/C4.[15]

Victor Kac noticed that all nite order automorphisms of a simple Lie algebra can be determined using the corre-sponding ane Lie algebra: that classication, which leads to an alternative method of classifying pairs ( g , ), isdescribed in Helgason (1978).

20 CHAPTER 9. BORELDE SIEBENTHAL THEORY

9.5 Applications to hermitian symmetric spaces of compact typeThe equal rank case with K non-semisimple corresponds exactly to the Hermitian symmetric spacesG / K of compacttype.In fact the symmetric space has an almost complex structure preserving the Riemannian metric if and only if there isa linear map J with J2 = I on p which preserves the inner product and commutes with the action of K. In this case Jlies in k and exp Jt forms a one-parameter group in the center of K. This follows because if A, B, C, D lie in p , thenby the invariance of the inner product on g [16]

([[A;B]; C]; D) = ([A;B]; [C;D]) = ([[C;D]; B]; A):

Replacing A and B by JA and JB, it follows that

[JA; JB] = [A;B]:

Dene a linear map on g by extending J to be 0 on k . The last relation shows that is a derivation of g . Since gis semisimple, must be an inner derivation, so that

(X) = [T +A;X];

with T in k and A in p . Taking X in k , it follows that A = 0 and T lies in the center of k and hence that K isnon-semisimple. [17]

If on the other hand G / K is irreducible with K non-semisimple, the compact group G must be simple and K ofmaximal rank. From the theorem of Borel and de Siebenthal, the involution is inner and K is the centralizer of atorus S. It follows that G / K is simply connected and there is a parabolic subgroup P in the complexication GC ofG such that G / K = GC / P. In particular there is a complex structure on G / K and the action of G is holomorphic.In general any compact hermitian symmetric space is simply connected and can be written as a direct product ofirreducible hermitian symmetric spaces Gi / Ki with Gi simple. The irreducible ones are exactly the non-semisimplecases described above.[18]

9.6 Notes[1] Helgason 1978

[2] Wolf 2010