Involutions of groups with twin BN -pair
Max Horn
February 2, 2009
Abstract
Let G be a group with a (saturated) twin BN -pair (B+ , B− , N ),
i.e., subgroups B+ , B− , N of G satisfying certain conditions, in partic-
ular: G = B+ , N = B+ , N ; the subgroup T := B+ ∩ B− satisfies
T = B+ ∩ N = B− ∩ N N and W := N/T is a Coxeter group. Ex-
amples include connected reductive algebraic groups and Kac-Moody
groups.
We study properties of involutory automorphisms θ of G satis-
fying that θ(B+ ) is conjugate to B− . Such group automorphisms
closely correspond to certain automorphisms of the twin building C =
(G/B+ , G/B− ) associated to G, namely those which interchange the
halves of the twin building almost isometrically.
Using this geometric correspondence we derive results on proper-
ties of G and the centralizer of θ in G. As an example, let G be
SLn (C), let θ be the twisted Chevalley involution (i.e., x → T x−1 ,¯
·
where ¯ is complex conjugation). Then we can e.g. derive finiteness
properties of SUn = CG (θ). This generalizes to arbitrary reductive
algebraic group or Kac-Moody group over arbitrary fields.
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