Synopses & Reviews
Descent in Buildings begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. The authors then put their algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving necessary and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or "form" of the original building. At the center of this theory is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. These results are combined at the end to show that every exceptional Bruhat-Tits building arises as a form of a "residually pseudo-split" Bruhat-Tits building. The book concludes with a display of the Tits indices associated with each of these exceptional forms.
This is the third and final volume of a trilogy that began with Richard Weiss' The Structure of Spherical Buildings and The Structure of Affine Buildings.
Synopsis
The Description for this book, Descent in Buildings (AMS-190), will be forthcoming.
About the Author
Bernhard Mühlherr is professor of mathematics at the University of Giessen in Germany. Holger P. Petersson is professor emeritus of mathematics at the University of Hagen in Germany. Richard M. Weiss is the William Walker Professor of Mathematics at Tufts University. He is the author of The Structure of Spherical Buildings, Quadrangular Algebras and The Structure of Affine Buildings (all Princeton) and the coauthor with Jacques Tits of Moufang Polygons.
Table of Contents
Preface xi
PART 1. MOUFANG QUADRANGLES 1
Chapter 1. Buildings 3
Chapter 2. Quadratic Forms 13
Chapter 3. Moufang Polygons 23
Chapter 4. Moufang Quadrangles 31
Chapter 5. Linked Tori, I 41
Chapter 6. Linked Tori, II 47
Chapter 7. Quadratic Forms over a Local Field 57
Chapter 8. Quadratic Forms of Type E6, E7 and E8 69
Chapter 9. Quadratic Forms of Type F4 79
PART 2. RESIDUES IN BRUHAT-TITS BUILDINGS 83
Chapter 10. Residues 85
Chapter 11. Unramified Quadrangles of Type E6, E7 and E8 91
Chapter 12. Semi-ramified Quadrangles of Type E6, E7 and E8 93
Chapter 13. Ramified Quadrangles of Type E6, E7 and E8 101
Chapter 14. Quadrangles of Type E6, E7 and E8: Summary 109
Chapter 15. Totally Wild Quadratic Forms of Type E7 115
Chapter 16. Existence 119
Chapter 17. Quadrangles of Type F4 129
Chapter 18. The Other Bruhat-Tits Buildings 137
PART 3. DESCENT 141
Chapter 19. Coxeter Groups 143
Chapter 20. Tits Indices 153
Chapter 21. Parallel Residues 165
Chapter 22. Fixed Point Buildings 181
Chapter 23. Subbuildings 195
Chapter 24. Moufang Structures 205
Chapter 25. Fixed Apartments 217
Chapter 26. The Standard Metric 221
Chapter 27. Affine Fixed Point Buildings 233
PART 4. GALOIS INVOLUTIONS 241
Chapter 28. Pseudo-Split Buildings 243
Chapter 29. Linear Automorphisms 251
Chapter 30. Strictly Semi-linear Automorphisms 259
Chapter 31. Galois Involutions 271
Chapter 32. Unramified Galois Involutions 275
PART 5. EXCEPTIONAL TITS INDICES 285
Chapter 33. Residually Pseudo-Split Buildings 287
Chapter 34. Forms of Residually Pseudo-Split Buildings 297
Chapter 35. Orthogonal Buildings 303
Chapter 36. Indices for the Exceptional Bruhat-Tits Buildings 309
Bibliography 327
Index 333