Involutions of Kac-Moody groups

Abstract

Goal of the present work is the study of involutory automorphisms and their centralizers of reductive algebraic groups and of split Kac-Moody groups, outside of characteristic 2. The groups in question have in common that they admit a so-called twin BN-pair (B_+, B_-, N) and an associated so-called twin building C=(C_+, C_-, δ^*). Let G be such a group. An involutory automorphism θ of G for which θ(B_+) is conjugate to B_- induces an almost isometry of the building C which interchanges the halves of the building and which we also denote by θ. This now enables us to apply the rich structure theory of buildings. An important tool for this is the so-called flip-flop system C_θ, consisting of all chambers c of C_+ for which the distance between c and θ(c) is maximal (with respect to the codistance of the twin building). Since C_θ is a subsystem of the building C_+, we can also regard it as a simplicial complex. The centralizer G_θ of θ in G acts naturally on this complex. In the present work we give criteria under which C_θ is a connected and pure simplicial complex. For this we reduce the global question to a problem in rank 2. We then solve this rank 2 problem for several important cases. Additional, we study the orbit structure of G_Simplizialkomplex on the building C and the flip-flop system C_θ. As applications, we obtain for example a parametrization of the double coset space G_θ\G/B_+; a generlization of the Iwasawa decomposition; and a proof that for certain locally-finite Kac-Moody groups, the centralizer C_θ is finitely generated. In closing we would like to remark that our results also apply to further groups with a root group datum as defined by Tits (like e.g. finite groups of Lie-type).