Buildings of Spherical Type and Finite BN-Pairs

1. Front Matter

   Pages I-X

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2. Complexes

   Pages 1-12

3. Coxeter complexes

   Pages 13-37

4. Buildings

   Pages 38-59

5. Reduction

   Pages 60-74

6. The building of a semi-simple algebraic group

   Pages 75-91

7. Buildings of type An, Dn, En

   Pages 92-101

8. Buildings of type Cn. I. Polar spaces

   Pages 102-116

9. Buildings of type Cn. II. Projective embeddings of polar spaces

   Pages 117-176

10. Buildings of type Cn. III. Non-embeddable polar spaces

    Pages 177-199

11. Buildings of type F4

    Pages 200-217

12. Finite BN-pairs of irreducible type and rank ≥ 3

    Pages 218-231

13. Appendix 1. Shadows

    Pages 232-248

14. Appendix 2. Generators and relations

    Pages 249-273

15. Back Matter

    Pages 274-302

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These notes are a slightly revised and extended version of mim- graphed notes written on the occasion of a seminar on buildings and BN-pairs held at Oberwolfach in April 1968. Their main purpose is to present the solution of the following two problems: (A) Determination of the buildings of rank >; and irreducible, spherical type, other than ~ and H ("of spherical type" means "with finite Weyl 4 group", about the excluded types H, cf. the addenda on p. 274). Roughly speaking, those buildings all turn out to be associated to simple algebraic or classical groups (cf. 6. ;, 6. 1;, 8. 4. ;, 8. 22, 9. 1, 10. 2). An easy application provides the enumeration of all finite groups with BN-pairs of irreducible type and rank >;, up to normal subgroups contained in B (cf. 11. 7). (B) Determination of all isomorphisms between buildings of rank > 2 and spherical type associated to algebraic or classical simple groups and, in parti­ cular, description of the full automorphism groups of such buildings (cf. 5. 8, 5. 9, 5. 10, 6. 6, 6. 1;, 8. 6, 9. ;, 10. 4). Except for the appendices, the notes are rather strictly oriented - ward these goals.

• Finite
• Morphism
• algebra
• algebraic group
• automorphism
• finite group

• Collège de France, Paris Cedex 05, France

  Jacques Tits