Buildings of Spherical Type and Finite BN-Pairs
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.
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adjacence-preserving algebraic group apartment containing assertion assume automorphism belong bijection BN-pair building of type called Cayley algebra Cham St chamber complex codimension coincide collinear conformal space convex set COROLLARY Coxeter complex cp(A cp(C cp(D cp(S cr,e defined denote diagram dist division ring Dynkin diagram embedding endomorphism finite fixed flag complex form f full convex hull group G group of type hence homomorphism immediate consequence implies incident induces intersection isomorphism leaves invariant LEMMA let cp Let G linear subspaces mapping maximal subspaces morphism notations opposite pair parabolic subgroups plane points polar space polar space associated proj proJA projective line projective space proof properties PROPOSITION prove quadratic form resp restriction roots satisfied sesquilinear form simple groups space of rank subcomplex subgroup of G subset Sylow subgroups symplecta theorem vector space vertices of diagr weak building Weyl group