Edifices: Building-like Spaces Associated to Linear Algebraic Groups; In memory of Jacques Tits

Abstract

Given a semisimple linear algebraic k-group G, one has a spherical building ΔG, and one can interpret the geometric realisation ΔG(R) of ΔG in terms of cocharacters of G. The aim of this paper is to extend this construction to the case when G is an arbitrary connected linear algebraic group; we call the resulting object ΔG(R) the spherical edifice of G. We also define an object VG(R) which is an analogue of the vector building for a semisimple group; we call VG(R) the vector edifice. The notions of a linear map and an isomorphism between edifices are introduced; we construct some linear maps arising from natural group-theoretic operations. We also devise a family of metrics on VG(R) and show they are all bi-Lipschitz equivalent to each other; with this extra structure, VG(R) becomes a complete metric space. Finally, we present some motivation in terms of geometric invariant theory and variations on the Tits Centre Conjecture.