Parabolic subalgebras, parabolic buildings and parabolic projection

Abstract: Reductive (or semisimple) algebraic groups, Lie groups and Lie algebras have a rich geometry determined by their parabolic subgroups and subalgebras, which carry the structure of a building in the sense of J. Tits. We present herein an elementary approach to the geometry of parabolic subalgebras, over an arbitrary field of characteristic zero, which does not rely upon the structure theory of semisimple Lie algebras. Indeed we derive such structure theory, from root systems to the Bruhat decomposition, from the properties of parabolic subalgebras. As well as constructing the Tits building of a reductive Lie algebra, we establish a "parabolic projection" process which sends parabolic subalgebras of a reductive Lie algebra to parabolic subalgebras of a Levi subquotient. We indicate how these ideas may be used to study geometric configurations and their moduli.