Involutions, Classical Groups, and Buildings

In [Invent. Math.58 (1980), 201–210], Curtis et al. construct a variation of the Tits building. The Curtis–Lehrer–Tits building ( , k) of a connected reductive k-group has the important feature that it is a functor from the category of reductive groups defined over a field k and monomorphisms to the category of topological spaces and inclusions. An important consequence derived by Curtis et al. from the functorial nature of the Curtis–Lehrer–Tits building ( , k) is that if s is a semisimple element of the group (k) of k-rational points, and ′ is the connected component group of the centralizer of s, then the fixed point set ( , k)s of s in ( , k) is the Curtis–Lehrer–Tits building ( ′, k). We generalize this result to arbitrary involutions of Autk( ), and we also prove an analogue in the context of affine buildings.