Tits' systems with crystallographic Weyl groups

Abstract

A Coxeter group is a group W with a set of generators w, ,..., w! and a presentation of the form (zuiwJm”j = 1, i, j E { 1,. . . , d}, where each mii is either co or a positive integer satisfying mii = 1, mij = mji 2 2 if i + j (the symbol (wiwJ” = 1 is to be interpreted as meaning that this relation may be omitted from the presentation). We will say that W is crystallographic if it has such a presentation and each mi, = 2, 3,4, 6, or, co. In this paper, we prove that any crystallographic Coxeter group appears as the Weyl group of some Tits’ system (or (B, N)-pair) [I 1 by 1, Chap. IV, Section 21. The Tits’ systems in question arise as automorphism groups of certain Lie algebras generalizing the classical semisimple Lie algebras. The construction is valid over any field of characteristic 0 or, with minor restrictions on the characteristic, any field of characteristic p > 0 which appears as the residue class field of some field of characteristic 0 with a nonarchimedean valuation. In the first case, the automorphism groups are just the natural generalization of the adjoint Chevalley groups. In the second case, they appear as groups induced by p-adic Chevalley groups. The essential difficulty is the lack of any generalization of Chevalley’s commutator formula. We have circumvented this issue by defining the Bore1 groups differently though the definitions coincide in the classical cases.